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1 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I: CONSTRUCTION OF HODGE THEATERS Shinichi Mochizuki February 2019 The present paper is the first in a series of four papers, the Abstract. Teichm ̈ uller theory for number arithmetic version of goal of which is to establish an —whichwerefertoas “inter-universal elliptic curve fields equipped with an — by applying the theory of semi-graphs of anabelioids , Teichm ̈ uller theory” ́ Frobenioids , and log-shells developed in earlier papers by ,the etale theta function “initial -data” , which consists of the author. We begin by fixing what we call Θ an elliptic curve E over a number field F , and a 5, as well as l ≥ prime number F some other technical data satisfying certain technical properties. This data deter- mines various etale coverings to the hyperbolic orbicurves that are related via finite ́ X once-punctured elliptic curve E . These finite ́ etale coverings determined by F F symmetry properties arising from the additive and multiplicative admit various of the elliptic curve. = Z /l Z acting on the l -torsion points F structures on the ring l ± ell NF-Hodge theaters” associated to the given Θ-data. These We then construct “ Θ ell ± Θ NF-Hodge theaters may be thought of as conventional miniature models of in which the scheme theory two underlying combinatorial dimensions of a and additive number field — which may be thought of as corresponding to the multiplicative group of units and structures of a ring or, alternatively, to the value group of a local field associated to the number field — are, in some sense, ell ± NF-Hodge theaters from one another. All Θ or “dismantled” “disentangled” are isomorphic to one another, but may also be related to one another by means of a ell ± , which relates certain Frobenioid-theoretic portions of one Θ Θ “ -link” NF-Hodge theater to another in a fashion that is with the respective conven- not compatible tional ring/scheme theory structures . In particular, it is a highly nontrivial problem to relate the ring structures on either side of the Θ-link to one another. This will be achieved, up to certain “relatively mild indeterminacies” , in future papers in the series by applying the absolute anabelian geometry developed in earlier description of an [asso- papers by the author. The resulting “alien ring structure” ciated, say, to the domain of the Θ-link] in terms of a given ring structure [associated, codomain say, to the of the Θ-link] will be applied in the final paper of the series to obtain results in diophantine geometry . Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tem- pered fundamental group of a p -adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest. Contents: Introduction § 0. Notations and Conventions § 1. Complements on Coverings of Punctured Elliptic Curves Typeset by A X S -T E M 1

2 2 SHINICHI MOCHIZUKI 2. Complements on Tempered Coverings § § 3. Chains of Θ-Hodge Theaters uller Theory § 4. Multiplicative Combinatorial Teichm ̈ 5. ΘNF-Hodge Theaters § 6. Additive Combinatorial Teichm ̈ uller Theory § Introduction I1. Summary of Main Results § § I2. Gluing Together Models of Conventional Scheme Theory I3. Basepoints and Inter-universality § p § uller Theory I4. Relation to Complex and -adic Teichm ̈ I5. Other Galois-theoretic Approaches to Diophantine Geometry § Acknowledgements I1. Summary of Main Results § The present paper is the first in a series of four papers, the goal of which is version of to establish an uller theory for number fields arithmetic Teichm ̈ elliptic curve , by applying the theory of equipped with an semi-graphs of anabe- lioids Frobenioids ,the ́ etale theta function ,and log-shells developedin[SemiAnbd], , [FrdI], [FrdII], [EtTh], and [AbsTopIII] [cf., especially, [EtTh] and [AbsTopIII]]. Unlike many mathematical papers, which are devoted to verifying properties of mathematical objects that are either well-known or easily constructed from well- known mathematical objects, in the present series of papers, most of our efforts will be devoted to constructing new mathematical objects . It is only in the final portion of the third paper in the series, i.e., [IUTchIII], that we turn to the task of concerning the mathematical objects constructed. In proving properties of interest the fourth paper of the series, i.e., [IUTchIV], we show that these properties may be combined with certain elementary computations to obtain diophantine results concerning elliptic curves over number fields. § 0 below for more on the notations and We refer to appliedinthe conventions present series of papers. The starting point of our constructions is a collection of initial Θ -data [cf. Definition 3.1]. Roughly speaking, this data consists, essentially, of , F number field over a E an · elliptic curve F of F , F an · algebraic closure prime number l ≥ 5, · a F ⊆ K ,and of a certain subfield V valuations a · collection of bad of a certain subfield F ⊆ F collection of · V valuations a mod mod that satisfy certain technical conditions — we refer to Definition 3.1 for more details. for the extension field F ⊆ ⊆ F for the field of moduli of E K , F Here, we write F mod once-punctured for the E ⊆ X , E -torsion points l determined by the F of of F F F C → for the ,and X E obtained by removing the origin from elliptic curve F F F by the hyperbolic orbicurve obtained by forming the stack-theoretic quotient of X F

3 ̈ INTER-UNIVERSAL TEICHM 3 ULLER THEORY I {± natural action of .Then F is assumed to be Galois over F 1 , Gal( K/F ) } mod E F )that contains SL ( ( F ), GL is assumed to be isomorphic to a subgroup of l F 2 l 2 at all of the nonarchimedean valuations of , stable reduction is assumed to have F def C C K × V K is assumed to be a = -core [cf. [CanLift], Remark 2.1.1], F F K such that the natural inclusion K is assumed to be a collection of valuations of ∼ ⊆ ⊆ K induces a bijection V V → V F between V and the set of all F mod mod mod ,and F valuations of the number field mod bad V V ⊆ mod mod is assumed to be some of nonarchimedean valuations of odd residue nonempty set — i.e., roughly bad [i.e., multiplicative] reduction has characteristic over which E F has bad multiplicative reduc- of the set of valuations where E speaking, the subset F tion that will be “of interest” to us in the context of the theory of the present series def def good bad bad bad of papers. Then we shall write V V V ⊆ V , V = × , = V V \ mod V mod mod mod mod def good bad V \ V . Also, we shall apply the superscripts “non” and “arc” to = V , V V mod nonarchimedean to denote the subsets of valuations, respectively. and archimedean -isomorphism [cf. Remark 3.1.3], a etale K This data determines, up to finite ́ covering C → of degree l such that the base-changed covering C K K def def X K × × C X = → = X X K F F C F K K F ∼ l ]of l [ l ]  Q ( ) of the module [ Z /l Z - E arises from a E rank one quotient = K K def bad v E , ( = E )[wherewewrite × V K ]which,at K ∈ torsion points of E K F K F coverings of the dual graph of the special fiber restricts to the quotient arising from . Moreover, the above data also determines a cusp  bad ,upto ∈ V C , corresponds to the canonical generator v ± 1, of Q which, at of K of the dual graph of the special [i.e., the generator determined by the unique loop bad ∈ , one obtains a natural finite ́ etale covering of V v fiber]. Furthermore, at degree l def def X K ) = X × × C K = ( → C → X K v K v v K K v v good ∈ V , one obtains a natural -th roots of the theta function; at v by extracting l l etale covering of degree finite ́ def def ) K X = × = X C × C K → ( → X v v K K v K v K → − v X . More details on the structure of the coverings , X , C [for  determined by K K v bad good v ], X ∈ 1ofthe § [for v V V ∈ ] may be found in [EtTh], § 2, as well as in − → v present paper. In this situation, the objects def def def def   ± ×  ± l } =( l +1) / 2; F − 1 1) = F / {± / {± 1 } ; F 2; l  =( = F l l l l l

4 4 SHINICHI MOCHIZUKI 4; Definitions 6.1, 6.4] will play an important § [cf. the discussion at the beginning of role in the discussion to follow. The natural action of the stabilizer in Gal( K/F )of  of  Q on Q determines a natural poly-action [ F l ] on C , i.e., the quotient E K K l  a natural isomorphism of F with some of Aut( C ) [cf. Example 4.3, subquotient K l   constituted by this poly-action of -symmetry may be thought F F (iv)]. The l l of as being essentially arithmetic in nature, in the sense that the subquotient of  F ) that gives rise to this poly-action of is induced, via the natural map Aut( C K l → Aut( K ), by a subquotient of Gal( ) ) ⊆ Aut( K ). In a similar vein, K/F Aut( C K X on the cusps of the natural action of the automorphisms of the scheme X K K  ±  ± natural poly-action on of F , i.e., a natural isomorphism of F X determines a K l l  ± X with some ) [cf. Definition 6.1, (v)]. The F subquotient of Aut( -symmetry K l  ± may be thought of as being essentially geo- constituted by this poly-action of F l X X ) ⊆ Aut( ( ) [i.e., of metric in nature, in the sense that the subgroup Aut K K K ) onto the subquotient of Aut( K -linear automorphisms] maps isomorphically X K  ±  . On the other hand, the - F global F that gives rise to this poly-action of l l -symmetry” [i.e., in essence, fails to extend!] } 1 only extends to a “ { of symmetry C K good bad ∈ V [for v ∈ V global ]and X ], while the v [for X coverings local of the − → v v  ± -symmetry” [i.e., in essence, fails to -symmetry of X } only extends to a “ {± 1 F K l good bad V ∈ v ] — cf. Fig. v ∈ V [for ]and X [for X coverings local extend!] of the → − v v I1.1 below. {± 1 } X X {  } or v V ∈ → − v v  ±   F F l l C X   K K Fig. I1.1: Symmetries of coverings of X F bad for the We shall write Π of X V tempered fundamental group v ∈ ,when v v etale fundamental group of X ́ , for the [cf. Definition 3.1, (e)]; we shall write Π v − → v good non v when ∈ v V V [cf. Definition 3.1, (f)]. Also, for , we shall write Π G  ∈ v v absolute Galois group of the base field for the quotient determined by the .Often, K v in the present series of papers, we shall consider various types of collections of data ∼ V → V ∈ )that ( v — indexed by — which we shall refer to as “prime-strips” mod bad ]or v ∈ V [when X are isomorphic to certain data that arise naturally from X − → v v good ∈ V [when ]. The main types of prime-strips that will be considered in the v present series of papers are summarized in Fig. I1.2 below. Perhaps the most basic kind of prime-strip is a -prime-strip .When v D ∈ non ,theportionofa D -prime-strip labeled by v is given by a category equivalent V to [the full subcategory determined by the connected objects of] the category of bad X of etale coverings [when v ∈ V [when ]or finite ́ X of tempered coverings − → v v good arc v ]. When ∈ ∈ V V , an analogous definition may be obtained by applying v the theory of Aut -holomorphic orbispaces developed in [AbsTopIII], § 2. One variant non  V -prime-strip .When v ∈ , -prime-strip is the notion of a D D of the notion of a  -prime-strip labeled by v is given by a category equivalent to the portion of a D [the full subcategory determined by the connected objects of] the Galois category

5 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 5 arc associated to ∈ V , an analogous definition may be given. In some ;when G v v “local arithmetic -prime-strips may be thought of as abstractions of the sense, D [which we regard as equipped with of [copies of] F holomorphic structure” mod ] — cf. the discussion of [AbsTopIII], § I3. On X the once-punctured elliptic curve F  -prime-strips may be thought of as “mono-analyticizations” the other hand, D [i.e., roughly speaking, the arithmetic version of the underlying real analytic struc- -prime-strips — cf. the discussion of ture associated to a holomorphic structure] of D I3. Throughout the present series of papers, we shall use the notation [AbsTopIII], §  mono-analytic to denote structures. Next, we recall the notion of a Frobenioid over a base category [cf. [FrdI] [typically denoted “ F ”] may for more details]. Roughly speaking, a Frobenioid be thought of as a category-theoretic abstraction of the notion of a category of base category D ”] line bundles or monoids of divisors over a [typically denoted “ “topos” Galois cate- of topological localizations [i.e., in the spirit of a ]suchasa  -prime-strips, we shall also consider various types -and D gory . In addition to D of prime-strips that arise from considering various natural Frobenioids — i.e., more ∈ V .Per- v concretely, various natural monoids equipped with a Galois action —at haps the most basic type of prime-strip arising from such a natural monoid is an bad determine, ∈ V .Then F v and F -prime-strip v . Suppose, for simplicity, that F K .Write of algebraic closure up to conjugacy, an v v ·O ; for the ring of integers of F v F v  for the multiplicative monoid of nonzero integers; ⊆O ·O F v F v × for the multiplicative monoid of units; ⊆O ·O F v F v μ ·O for the multiplicative monoid of roots of unity; ⊆O F v F v μ l 2 l for the multiplicative monoid of 2 -th roots of unity; ⊆O ·O F v F v · q v . for a 2 l -th root of the q -parameter of E ∈O at F F v v μ μ ×  l 2 -actions , O , O G .The O O natural are equipped with ,and , O Thus, v F v F F F F v v v v is given by data isomorphic to the monoid F -prime-strip labeled by v portion of an   , equipped with its natural Π )-action [cf. Fig. I1.2]. There are various ( G O v v F v versions of the notion of an F -prime-strip; perhaps the most basic mono-analytic   -prime-strip .Theportionofan F -prime-strip labeled by F is the notion of an × N , equipped with its natural O × q is given by data isomorphic to the monoid v F v v G -action [cf. Fig. I1.2]. Often we shall regard these various mono-analytic ver- v F -prime-strip as being equipped with an additional global realified sions of an , which, at a concrete level, corresponds, essentially, to considering var- Frobenioid ∼ ∈ V ( → V ) that are related to one another by arithmetic degrees ∈ R at v ious mod means of the product formula . Throughout the present series of papers, we shall use the notation 

6 6 SHINICHI MOCHIZUKI to denote such prime-strips. bad v V Reference Model at ∈ Type of prime-strip I, 4.1, (i) D Π v  G I, 4.1, (iii) D v  Π  O I, 5.2, (i) F v F v × N   O F × q G I, 5.2, (ii) v F v v × ×  O G F II, 4.9, (vii) v F v def × μ μ × × μ F II, 4.9, (vii)  O = O G O / v F F F v v v μ ×  × μ N   × q F G II, 4.9, (vii) O v F v v   N F  q G III, 2.4, (ii) v v μ 2 l ⊥ N O III, 2.4, (ii) G F  q × v F v v } {  ...  ... = F global realified Frobenioid associated to F + F mod Fig. I1.2: Types of prime-strips In some sense, the main goal of the present paper may be thought of as the ± ell construction of Θ NF-Hodge theaters [cf. Definition 6.13, (i)] ± ell Θ NF † HT — which may be thought of as “miniature models of conventional scheme the- ory ” — given, roughly speaking, by systems of Frobenioids .Toanysuch

7 ̈ INTER-UNIVERSAL TEICHM 7 ULLER THEORY I ± ell Θ NF ± † ± ell ell D - Θ Θ HT NF-Hodge the- , one may associate a NF-Hodge theater ater [cf. Definition 6.13, (ii)] ± ell -Θ NF D † HT . system of base categories — i.e., the associated ± ell Θ NF ell † ± NF-Hodge theater as the result of gluing HT One may think of a Θ ± ell ΘNF Θ † † ± ell ΘNF -Hodge theater HT HT -Hodge theater [cf. Re- to a together a Θ ± ell NF-Hodge theater D -Θ mark 6.12.2, (ii)]. In a similar vein, one may think of a ± ell ell ± -Θ D NF -Θ D ± ell † † -Θ HT HT as the result of gluing together a D -Hodge theater ell ± D D -ΘNF -Θ † ± ell † -Hodge theater .A HT HT D -ΘNF-Hodge theater D may to a -Θ that allows one to keep track of the action bookkeeping device be thought of as a  ± on the labels -symmetry of the F l   <...< ) − 1 < 0 < 1 <...

8 8 SHINICHI MOCHIZUKI appear in the present series of papers. Perhaps most importantly, the theory of the log-Frobenius functor § 3, § 4, § 5, was in- and log-shells developed in [AbsTopIII], tended as a prototype for the theory of the log -link that is developed in [IUTchIII]. In particular, although most of the main of [AbsTopIII], ideas and techniques 4, § 5, will play an important role in the present series of papers, many of the 3, § § 3, § 4, § 5, will not be applied in a direct , constructions performed in [AbsTopIII], § sense in the present series of papers. literal ±  -symmetry has the advantange that, being geometric in nature, it The F l non ”[where v ∈ V ] associated to dis- allows one to permute various copies of “ G v . This phenomenon, without inducing conjugacy indeterminacies labels tinct F ∈ l conjugate synchronization which we shall refer to as , will play a key role in surrounding the the Kummer theory Hodge-Arakelov-theoretic evaluation of the l -torsion points that is developed in [IUTchII]— cf. the dis- theta function at cussion of Remark 6.12.6; [IUTchII], Remark 3.5.2, (ii), (iii); [IUTchII], Remark  -symmetry is more suited to situations in which one F 4.5.3, (i). By contrast, the l . In the present series of papers, the most important from to F descend must K mod Kummer theory surrounding the reconstruction of such situation involves the etale fundamental group of from the ́ C —cf. thedis- the F number field mod K cussion of Remark 6.12.6; [IUTchII], Remark 4.7.6. This reconstruction will be discussed in Example 5.1 of the present paper. Here, we note that such situations ”[where of the various copies of “ necessarily induce global Galois permutations G v non  v onlywell-defineduptocon- ] associated to distinct labels ∈ F ∈ V that are l  -symmetry is ill-suited to situations, jugacy indeterminacies . In particular, the F l such as those that appear in the theory of that Hodge-Arakelov-theoretic evaluation is developed in [IUTchII], that require one to establish conjugate synchronization . ( ] [ ) ( )  l − 1 < 0 < ... < − <... 1 <... 1 1 {± }  ⇐ ⇒    <...

9 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 9 I3], of the ring — cf. the discussion of Remarks the discussion of [AbsTopIII], § Z 6.12.3, 6.12.6. Alternatively, this decomposition into additive and multiplicative symmetries ± ell NF-Hodge theaters may be compared to groups of addi- in the theory of Θ and of the upper half-plane [cf. Fig. I1.4 tive multiplicative symmetries geometry expressed by the additive symmetries of below]. Here, the “cuspidal” the upper half-plane admits a natural “associated coordinate”, namely, the clas-  ± -symmetry q , which is reminiscent of the way in which the F sical -parameter l Kummer theory Hodge-Arakelov-theoretic surrounding the is well-adapted to the theta function l -torsion points [cf. the above discussion]. evaluation of the at “toral” ,or “nodal” [cf. the classical theory of the structure of By contrast, the modulo Hecke correspondences p ], geometry expressed by the multiplicative sym- metries of the upper half-plane admits a natural “associated coordinate”, namely, unit disc , the classical biholomorphic isomorphism of the upper half-plane with the  -symmetry is well-adapted to the F which is reminiscent of the way in which the l [cf. the above discussion]. Kummer theory number field F surrounding the mod For more details, we refer to the discussion of Remark 6.12.3, (iii). scheme-theoretic Hodge-Arakelov theory devel- From the point of view of the ell ± NF- combinatorial structure of a Θ oped in [HASurI], [HASurII], the theory of the Hodge theater — and, indeed, the theory of the present series of papers! — may be regarded as a sort of solution to the problem of constructing “global multiplicative sub- and “global canonical generators” [cf. the quotient “ Q ”and spaces” ” that appear in the above discussion!] cusp “  the —the nonexistence of which in a “naive, scheme-theoretic sense” constitutes the to applying the theory of [HASurI], [HASurII] to diophantine main obstruction may be geometry [cf. the discussion of Remark 4.3.1]. Indeed, prime-strips of the natural morphism Spec( ) → “local analytic sections” K thought of as ). Thus, it is precisely by working with such “local analytic sections” — F Spec( mod , as opposed to V i.e., more concretely, by working with the collection of valuations all valuations of K the set of “simulate” the notions — that one can, in some sense, of a or a “global canonical generator” . On the other “global multiplicative subspace” hand, such “simulated global objects” may only be achieved at the cost of “dismantling” “surgery” on, the global prime struc- ,orperforming ture involved [cf. the discussion of Remark 4.3.1] of the number fields — a quite drastic operation, which has the effect of precipitating numerous technical difficulties , whose resolution, via the theory of semi-graphs of anabelioids , Frobe- nioids ,the etale theta function ,and log-shells developed in [SemiAnbd], [FrdI], ́ [FrdII], [EtTh], and [AbsTopIII], constitutes the bulk of the theory of the present series of papers! From the point of view of “performing surgery on the global prime  that appear in the “arithmetic” labels structure of a number field”, the ∈ F l  F “miniature finite approxima- -symmetry may be thought of as a sort of l of this global prime structure , in the spirit of the idea of “Hodge theory at tion” finite resolution” discussed in [HASurI], § 1.3.4. On the other hand, the labels ∈ F l  ± that appear in the “geometric” F -symmetry may be thought of as a sort l

10 10 SHINICHI MOCHIZUKI “miniature finite approximation” of [i.e., of the natural tempered Z -coverings at E tempered coverings with Galois group ] of the Tate curves determined by Z F bad V , again in the spirit of the idea of “Hodge theory at finite resolution” ∈ v discussed in [HASurI], § 1.3.4. ell ± Classical Θ NF-Hodge theaters in inter-universal upper half-plane Teichm ̈ uller theory  ± z + a , F Additive z  - → l ∈ →− z + a ( a  R ) symmetry symmetry z def 2 πiz = e “Functions” assoc’d theta fn. evaluated at q to l -tors. [cf. I, 6.12.6, (ii)] add. symm. ± assoc’d V single Basepoint cusp at infinity add. symm. to [cf. I, 6.1, (v)] Combinatorial assoc’d cusp prototype cusp to add. symm. t · sin( z ) cos( t ) −  → Multiplicative -  F , z l )+cos( ) t sin( · z t ) t )+sin( t z · cos( t → z symmetry ) R ∈ symmetry (  t cos( − ) t sin( · ) z elements of the “Functions” def z − i assoc’d to = w F number field mod i + z mult. symm. [cf. I, 6.12.6, (iii)] ) ( ( ) t ) − sin( t ) ) )sin( t cos( t cos( ± un Bor   · F V = assoc’d V Basepoints F , l l sin( t t cos( − ) t sin( ) t )cos( ) mult. symm.  to [cf. I, 4.3, (i)] entire boundary of H } { p Combinatorial p nodes of mod of mod nodes prototype assoc’d Hecke correspondence Hecke correspondence to mult. symm. [cf. II, 4.11.4, (iii), (c)] [cf. II, 4.11.4, (iii), (c)]  ±  -, -symmetries F F Fig. I1.4: Comparison of l l with the geometry of the upper half-plane

11 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 11 bad  ∈ F v -prime- V As discussed above in our explanation of the models at for q of the elliptic curve E -th roots of the -parameters l strips, by considering the 2 F bad good v v ∈ V ∈ in such a way as to V at , and, roughly speaking, extending to   F -prime-strip “ ” F product formula satisfy the , one may construct a natural mod [cf. Example 3.5, (ii); Definition 5.2, (iv)]. This construction admits an abstract, algorithmic formulation “ Θ -Hodge that allows one to apply it to the underlying ± ell NF Θ  † ± ell HT so as to obtain an F NF-Hodge theater - arbitrary Θ theater” of an prime-strip  † F mod [cf. Definitions 3.6, (c); 5.2, (iv)]. On the other hand, by formally replacing the l -th roots of the q -parameters that appear in this construction by the reciprocal 2 l -th root of the Frobenioid-theoretic , which we shall denote of the theta function bad ”[for V v ], studied in [EtTh] [cf. also Example 3.2, (ii), of the present ∈ “Θ v abstract, algorithmic formulation for the construction of an paper], one obtains an  F -prime-strip †  F tht [cf. Definitions 3.6, (c); 5.2, (iv)] from [the underlying Θ-Hodge theater of] the ± ell Θ NF ± † ell HT NF-Hodge theater . Θ ± ell Θ NF ‡ ell ± Now let Θ HT NF-Hodge theater [relative to the given be another initial Θ-data]. Then we shall refer to the “full poly-isomorphism” of [i.e., the  -prime-strips F collection of all isomorphisms between] ∼   † ‡ → F F tht mod ell ± NF Θ † HT -link to [the under- from [the underlying Θ-Hodge theater of] as the Θ ± ell NF Θ ‡ lying Θ-Hodge theater of] HT [cf. Corollary 3.7, (i); Definition 5.2, (iv)]. One fundamental property of the Θ-link is the property that it induces a collection × -prime-strips of isomorphisms [in fact, the full poly-isomorphism] between the F ∼ × × † ‡ F F → mod mod †   ‡ F and F [cf. Corollary 3.7, (ii), (iii); [IUTchII], Definition associated to mod mod 4.9, (vii)]. ± ell Θ NF ± ell n { Now let NF-Hodge theaters HT } collection of distinct Θ be a Z n ∈ [relative to the given initial Θ-data] indexed by the integers. Thus, by applying the constructions just discussed, we obtain an infinite chain ± ell ell ell ± ± Θ Θ Θ Θ Θ NF NF Θ NF Θ n 1) ( n +1) − n ( HT HT HT −→ −→ −→ −→ ... ... ± ell [cf. Corollary 3.8], which will be re- NF-Hodge theaters -linked of Θ Θ Frobenius-picture [associated to the Θ-link]. One fundamen- ferred to as the tal property of this Frobenius-picture is the property that it fails to admit per- mutation automorphisms that switch adjacent indices n , n + 1, but leave the remaining indices Z fixed [cf. Corollary 3.8]. Roughly speaking, the Θ-link ∈ ell ± ell ± Θ NF Θ NF Θ n ( n +1) HT may be thought of as a HT formal correspondence −→ +1) n ( n →  Θ q v v

12 12 SHINICHI MOCHIZUKI [cf. Remark 3.8.1, (i)], which is depicted in Fig. I1.5 below. In fact, the Θ-link discussed in the present paper is only a simplified version of the “Θ-link” that will ultimately play a central role in the present series of papers. The construction of the version of the Θ-link that we shall ultimately be interested technically involved and, indeed, occupies the greater part of the theory in is quite to be developed in [IUTchII], [IUTchIII]. On the other hand, the simplified version discussed in the present paper is of interest in that it allows one to give a relatively straightforward introduction to many of the important qualitative properties of ́ etale-picture Frobenius-picture the Θ-link — such as the discussed above and the in the case central importance to be discussed below — that will continue to be of of the versions of the Θ-link that will be developed in [IUTchII], [IUTchIII]. ell ± ± ell NF Θ Θ NF n +1 n HT HT ---- ---- ---- n n ( +1) +1) n ( n Θ q  Θ  q ... ... v v v v ( n +1) n q → Θ  v v Fig. I1.5: Frobenius-picture associated to the Θ-link Now let us return to our discussion of the Frobenius-picture associated to the Θ- ×  † × D link. The F -prime-strip associated to the F may, in fact, be -prime-strip mod †   associated to a certain -prime-strip D F -prime- D naturally identified with the > † F [cf. the discussion preceding Example 5.4] that arises from the Θ-Hodge strip > ± ell Θ NF ± † ell NF-Hodge theater .The HT -prime-strip D theater underlying the Θ † † associated to the D F F -prime-strip is precisely the D -prime-strip depicted > >  ]” in Fig. I1.3. Thus, the Frobenius-picture discussed above <... > >   of D -prime-strips. That is to say, when regarded up to isomorphism, the D -  ( − ) D ” may be regarded as an invariant — i.e., a “mono-analytic prime-strip “ > ± ell NF-Hodge theaters that occur in the Frobenius-picture core” —ofthevariousΘ [cf. Corollaries 4.12, (ii); 6.10, (ii)]. Unlike the case with the Frobenius-picture, ell ± NF -Θ D ± ell n HT to this NF-Hodge theaters -Θ D relationships the of the various spokes in Fig. I1.6 below mono-analytic core — relationships that are depicted by arbitrary permutation symmetries among the spokes — are compatible with ± ell NF-Hodge theaters] — cf. Corollaries Z of the D -Θ [i.e., among the labels n ∈ 4.12, (iii); 6.10, (iii), (iv). The diagram depicted in Fig. I1.6 below will be referred to as the ́ etale-picture . Thus, the ́ etale-picture may, in some sense, be regarded as a collection of canonical splittings of the Frobenius-picture. The existence of such splittings suggests that

13 ̈ INTER-UNIVERSAL TEICHM 13 ULLER THEORY I absolute anabelian geometry by applying various results from to the D various tempered and ́ - etale fundamental groups that constitute each ell ± NF-Hodge theater in the ́ etale-picture, one may obtain algorithmic Θ descriptions of — i.e., roughly speaking, one may take a “glimpse” ell ± NF-Hodge the- of one Θ conventional scheme theory inside — the ± ell Θ NF m in terms of the conventional scheme theory associated HT ater ± ell Θ NF ± ell n HT NF-Hodge theater [i.e., where n  = m ]. to another Θ main themes Indeed, this point of view constitutes one of the of the theory developed in the present series of papers and will be of particular importance in our treatment in [IUTchIII] of the main results of the theory. ± ell NF D -Θ n HT ... ... | ± ± ell ell — — ( − )  NF -Θ NF D D -Θ 1 n +1 − n D HT HT > | ... ... ell ± D NF -Θ n +2 HT ± ell ́ NF-Hodge theaters Etale-picture of D Fig. I1.6: -Θ Frobenius-like — i.e., Before proceeding, we recall the “heuristic” notions of —and ́ etale-like — i.e., “indifferent to order” — mathematical “order-conscious” discussed in [FrdI], I4. These notions will play a key role in the theory § structures developed in the present series of papers. In particular, the terms “Frobenius- picture” and “ ́ etale-picture” introduced above are motivated by these notions. main result of the present paper may be summarized as follows. The  ±  ́ -Links, and Frobenius-/ -/ F Etale-Pic- Θ -Symmetries, F Theorem A. ( l l ± ell Fix a collection of initial Θ - NF-Hodge Theaters) Θ tures Associated to [cf. Definition 3.1], which determines, in particular, data ( E data , F, l, V ) as F ± ell [cf. NF-Hodge theater Θ in the above discussion. Then one may construct a Definition 6.13, (i)] ± ell Θ NF † HT Θ -data, as well as — in essence, a system of Frobenioids — associated to this initial ell ± NF D -Θ ± ell † HT NF-Hodge theater [cf. Definition 6.13, (ii)] Θ - an associated D

14 14 SHINICHI MOCHIZUKI — in essence, the system of base categories associated to the system of Frobenioids ± ell NF Θ † HT . ± ell NF Θ  ±  ± ell † (i) F ( The HT -and Θ F -Symmetries) NF-Hodge theater l l ell ± Θ ± ell † may be obtained as the result of -Hodge theater together a to gluing HT Θ ΘNF † -Hodge theater ΘNF a [cf. Remark 6.12.2, (ii)]; a similar statement holds HT ± ell NF D -Θ ell † ± ± ell - .The global portion of a D - Θ HT NF-Hodge theater - D for the Θ ± ell -Θ D † consists of a category equivalent to [the full subcategory HT Hodge theater etale coverings determined by the connected objects of] the Galois category of finite ́  ± X of the [orbi]curve F . This global portion is equipped with an -symmetry , K l  ± i.e., a poly-action by F on the labels l   ( l − 0 1 <...

15 ̈ INTER-UNIVERSAL TEICHM 15 ULLER THEORY I † ‡   associated to F F and [cf. Corollary 3.7, (ii), (iii); [IUTchII], Definition mod mod 4.9, (vii)]. ell ± NF Θ n ́ (iii) { Let (Frobenius-/ Etale-Pictures) } HT be a collection of ∈ n Z ± ell NF-Hodge theaters [relative to the given initial Θ -data] indexed distinct Θ by the integers. Then the infinite chain ± ell ± ell ± ell Θ Θ Θ Θ NF Θ NF NF Θ Θ 1) n ( n n ( − +1) −→ HT −→ HT −→ HT −→ ... ... ± ell NF-Hodge theaters will be referred to as the Frobenius- of Θ Θ -linked Θ -link] — cf. Fig. I1.5; Corollary 3.8. The Frobenius- picture [associated to the picture fails to admit permutation automorphisms that switch adjacent indices , n +1 , but leave the remaining indices ∈ Z fixed. The Frobenius-picture induces n full poly-isomorphisms an infinite chain of ∼ ∼ ∼ ∼ (  − n  1) ( n +1)  n → → → D D D ... → ... > > >    n D between the various -prime-strips , i.e., in essence, the D D -prime-strips > × n × associated to the F D - -prime-strips F . The relationships of the various mod ell ± D -Θ NF n ell ± HT to the “mono-analytic core” constituted NF-Hodge theaters Θ   ) ( − -prime-strip “ D ” regarded up to isomorphism — relationships that are by the D > depicted by in Fig. I1.6 — are compatible with arbitrary permutation spokes ± ell NF- Z of the D symmetries Θ among the spokes, i.e., among the labels n ∈ - Hodge theaters [cf. Corollaries 4.12, (ii), 6.10, (i)]. The diagram depicted in Fig. I1.6 will be referred to as the ́ etale-picture . In addition to the main result discussed above, we also prove a certain technical result concerning —cf. TheoremBbelow— tempered fundamental groups Hodge-Arakelov-theoretic that will be of use in our development of the theory of evaluation in [IUTchII]. This result is essentially a routine application of the the- of tempered fundamental groups developed in maximal compact subgroups ory of [SemiAnbd] [cf., especially, [SemiAnbd], Theorems 3.7, 5.4, as well as Remark 2.5.3, (ii), of the present paper]. Here, we recall that this theory of [SemiAnbd] may be thought of as a sort of “Combinatorial Section Conjecture” [cf. Remark 2.5.1 of the present paper; [IUTchII], Remark 1.12.4] — a point of view that is of particu- historical remarks lar interest in light of the § I5 below. Moreover, Theorem made in B is of interest in that independently of the theory of the present series of papers new proof of the it yields, for instance, a of the tempered fun- normal terminality damental group in its profinite completion, a result originally obtained in [Andr ́ e], Lemma 3.2.1, by means of other techniques [cf. Remark 2.4.1]. This new proof is of interest in that, unlike the techniques of [Andr ́ e], which are only available in the profinite case, this new proof [cf. Proposition 2.4, (iii)] holds in the case of ̂ ̂ ̂ -completions , for more general pro- Σ [i.e., not just the case of Σ Σ= Primes ]. Theorem B. (Profinite Conjugates of Tempered Decomposition and Inertia Groups) Let k be a mixed-characteristic [nonarchimedean] local over field X a hyperbolic curve , k . Write tp Π X

16 16 SHINICHI MOCHIZUKI tp tempered fundamental group π ( X ) [relative to a suitable basepoint] for the 1 ̂ X 4; [SemiAnbd], Example 3.10]; [cf. [Andr ́ Π § of e], etale fundamental ́ for the X . Thus, we have a group [relative to a suitable basepoint] of X natural inclusion tp ̂ Π ↪ → Π X X tp ̂ with the profinite completion of Π Π which allows one to identify . Then every X X ̂ ̂ decomposition group in Π inertia group in ) associated to Π (respectively, X X tp X )iscontainedin Π (respectively, to a cusp of a closed point or cusp of X if X tp tp (respectively, inertia group in ) Π and only if it is a decomposition group in Π X X (respectively, to a cusp of X ). Moreover, associated to a closed point or cusp of X tp tp ̂ (respectively, inertia Π -conjugate of Π contains a decomposition group in Π a X X X tp group in Π ) associated to a closed point or cusp of X (respectively, to a cusp of X tp X Π ) if and only if it is equal to . X Theorem B is [essentially] given as Corollary 2.5 [cf. also Remark 2.5.2] in § 2. Here, we note that although, in the statement of Corollary 2.5, the hyperbolic ,one k of stable reduction over the ring of integers O is assumed to admit X curve k verifies immediately [by applying Proposition 2.4, (iii)] that this assumption is, in . fact, unnecessary for the need to apply Theorem B important reason Finally, we remark that one ± ell in the context of the theory of Θ NF-Hodge theaters summarized in Theorem A  ± -symmetry , which will play a crucial role in the theory F is the following. The l of the present series of papers [cf., especially, [IUTchII], [IUTchIII]], depends, in an synchronization of the ± -indeterminacies that occur locally essential way, on the V [cf. Fig. I1.1]. Such a synchronization may only be obtained by ∈ at each v ± ell -Hodge theater under consideration. global portion making use of the of the Θ global ± -synchronizations On the other hand, in order to avail oneself of such  ± - [cf. Remark 6.12.4, (iii)], it is necessary to regard the various F of the labels l symmetry   <...

17 ̈ INTER-UNIVERSAL TEICHM 17 ULLER THEORY I “subsystems of Frobenioids” by means of the full poly-isomorphisms between the  -prime-strips constituted by certain F ∼  † ‡  → F F tht mod to form the Frobenius-picture . One fundamental observation in this context is the following: these gluing isomorphisms — i.e., in essence, the correspondences n +1) n ( → q  Θ v v outside lie — and hence the geometry of the resulting Frobenius-picture the framework of conventional scheme theory in the sense that they arise from ring homomorphisms ! do not ± ell Θ NF n HT of conventional scheme In particular, although each particular model theory is constructed within the framework of conventional scheme theory, the ± ell NF-Hodge relationship between the distinct [albeit abstractly isomorphic, as Θ theaters!] conventional scheme theories represented by, for instance, neighboring ± ell ± ell NF Θ NF Θ ± ell n +1 n HT HT NF-Hodge theaters cannot be expressed scheme- , Θ . In this context, it is also important to note that such gluing operations theoretically relatively simple structure — for instance, are possible precisely because of the by comparison to the structure of a ring ! — of the Frobenius-like structures  -prime-strips involved, F constituted by the Frobenioids that appear in the various [cf. Fig. I1.2]. monoids isomorphic to N i.e., in essence, collections of R or ≥ 0 ± ell etale-pictures of Θ Fig. I2.1: Depiction of Frobenius- and ́ NF-Hodge theaters via glued topological surfaces

18 18 SHINICHI MOCHIZUKI “conventional scheme theory” as being analo- If one thinks of the geometry of , then the geometry represented by the “Euclidean space” gous to the geometry of , i.e., which is obtained by Frobenius-picture corresponds to a “topological manifold” not homeomorphic gluing together various portions of Euclidean space, but which is to Euclidean space. This point of view is illustrated in Fig. I2.1 above, where the ± ell Frobenius-picture NF-Hodge theaters two- in the are depicted as [ various Θ — cf. the discussion of § I1] twice-punctured topological surfaces dimensional! of genus one ,which , glued together along tubular neighborhoods of cycles one-dimensional § I1] mono-analytic correspond to the [ ! — cf. the discussion of . The permuta- -link data that appears in the isomorphism that constitutes the Θ § I1] are depicted in Fig. etale-picture [cf. the discussion of tion symmetries in the ́ [cf. the discussion of multiradiality in I2.1 as the anti-holomorphic reflection [IUTchII], Introduction!] around a gluing cycle between topological surfaces. spirit Another elementary example that illustrates the of the gluing operations be =0 , 1, let R i discussed in the present series of papers is the following. For i closed unit interval ⊆ R the [i.e., corresponding to real line acopyofthe ; I i i and ⊆ I for the Cantor set C [0 , 1] ⊆ R ]. Write 0 0 ∼ → I : φ C 0 1 for the arising from the Cantor function .Thenifonethinksof R and bijection 0 highly nontrivial ,thenitisa as being glued to one another by means of φ R 1 problem to describe structures naturally associated to the “alien” ring structure — such as, for instance, the subset of ∈ R — algebraic numbers of R 0 0 . R in terms that only require the use of the ring structure of 1 of the gluing op- A slightly less elementary example that illustrates the spirit erations discussed in the present series of papers is the following. This example is much closer to the theory of the present series of papers than the exam- technically ples involving topological surfaces and Cantor sets given above. For simplicity, let us write ×  ,G  O O G  ×   O for the pairs “ ” [cf. the notation of the discussion O ”, “ G  G v v F F v v surrounding Fig. I1.2]. Recall from [AbsTopIII], Proposition 3.2, (iv), that the operation  )  → G O (  G  O of “forgetting bijection from the group of automorphisms of the ” determines a  — i.e., thought of as an abstract ind-topological monoid equipped G  O pair with a continuous action by an abstract topological group — to the group of au- tomorphisms of the topological group G . By contrast, we recall from [AbsTopIII], Proposition 3.3, (ii), that the operation × G → )  G O  ( × O of “forgetting ” only determines a surjection from the group of automorphisms × — i.e., thought of as an abstract ind-topological monoid of the pair G  O

19 ̈ INTER-UNIVERSAL TEICHM 19 ULLER THEORY I equipped with a continuous action by an abstract topological group — to the group kernel G of automorphisms of the topological group ; that is to say, the of this × × ̂ . In particular, if one works on O natural action of Z surjection is given by the    , which one thinks of as being O O ,where i =0 , 1, of G  G with two copies i i glued to one another by means of an indeterminate isomorphism ∼ × × G ( ) )  O ( G →  O 1 0 1 0 × [i.e., where one thinks of each ( G ,  ), for i =0 O 1, as an abstract ind-topological i i monoid equipped with a continuous action by an abstract topological group], then, in general, it is a highly nontrivial problem   O )intermsthat structures naturally associated to ( G to describe 0 0  ).  O only require the use of ( G 1 1 One such structure which is of interest in the context of the present series of papers [cf., especially, the theory of [IUTchII], § 1] is the natural cyclotomic rigidity  between the group of torsion elements of O isomorphism and an analogous 0 — i.e., a structure that is G group of torsion elements naturally associated to 0 × × ̂ ! on O not preserved Z manifestly by the natural action of 0 In the context of the above discussion of Fig. I2.1, it is of interest to note the important role played by Kummer theory in the present series of papers [cf. the Introductions to [IUTchII], [IUTchIII]]. From the point of view of Fig. I2.1, this precise specification role corresponds to the of the gluing cycle within each twice- punctured genus one surface in the illustration. Of course, such a precise specifi- depends cation on the twice-punctured genus one surface under consideration, i.e., the same gluing cycle is subject to quite different “precise specifications” , relative left to the twice-punctured genus one surface on the and the twice-punctured genus one surface on the right . This state of affairs corresponds to the quite different Kummer theories to which the monoids/Frobenioids that appear in the Θ-link are ± ell of the Θ-link and NF-Hodge theater in the domain subject, relative to the Θ ell ± NF-Hodge theater in the codomain of the Θ-link. At first glance, it might the Θ Kummer theory appear that the use of , i.e., of the correspondence determined by Kummer classes , to achieve this precise specification of the relevant constructing ± ell NF-Hodge theater is somewhat arbitrary , monoids/Frobenioids within each Θ i.e., that one could perhaps use other correspondences [i.e., correspondences not determined by Kummer classes] to achieve such a precise specification. In fact, however, the rigidity of the relevant local and global monoids equipped with Ga- lois actions [cf. Corollary 5.3, (i), (ii), (iv)] implies that, if one imposes the natural ,then Galois-compatibility condition of the correspondence furnished by Kummer theory is the only accept- “ precise specification of the able choice for constructing the required ± ell NF-Hodge theater” Θ relevant monoids/Frobenioids within each — cf. also the discussion of [IUTchII], Remark 3.6.2, (ii). The construction of the Frobenius-picture described in § I1 is given in the present paper. More elaborate versions of this Frobenius-picture will be discussed in [IUTchII], [IUTchIII]. Once one constructs the Frobenius-picture, one natural

20 20 SHINICHI MOCHIZUKI , which will, in fact, be one of the main themes and fundamental problem of the present series of papers, is the problem of alien “arithmetic holomorphic structure” [i.e., an describing an ell ± Θ NF m HT alien “conventional scheme theory”] corresponding to some in terms of a “known arithmetic holomorphic structure” corresponding to ell ± Θ NF n n HT = m ] [where  § — a problem, which, as discussed in I1, will be approached, in the final portion of [IUTchIII], by applying various results from [i.e., absolute anabelian geometry more explicitly, the theory of [SemiAnbd], [EtTh], and [AbsTopIII]] to the various tempered and ́ etale-picture . etale fundamental groups that appear in the ́ The relevance to this problem of the extensive theory of “reconstruction of ring/scheme structures” provided by absolute anabelian geometry is evident from the statement of the problem. On the other hand, in this context, it is of interest to note that, unlike conventional anabelian geometry, which typically centers on the goal of reconstructing a “known scheme-theoretic object” , in the present series of papers, we wish to apply techniques and results from anabelian geometry in order to analyze the structure of an unknown , essentially object , non-scheme-theoretic namely, the Frobenius-picture , as described above. Put another way, relative to the point of view that “Galois groups are arithmetic tangent bundles” [cf. the theory of the arithmetic Kodaira-Spencer morphism in [HASurI]], one may think computation of the of conventional anabelian geometry as corresponding to the as automorphisms of a scheme 0 (arithmetic tangent bundle) H and of the application of absolute anabelian geometry to the analysis of the Frobenius- picture, i.e., to the solution of the problem discussed above, as corresponding to the computation of 1 (arithmetic tangent bundle) H computation of “deformations of the arithmetic holomorphic — i.e., the of a number field equipped with an elliptic curve. structure” “Hodge” in the In the context of the above discussion, we remark that the word term “Hodge theater” was intended as a reference to the use of the word “Hodge” in such classical terminology as “variation of Hodge structure ” [cf. also the discussion of Hodge filtrations § I5], for instance, in discussions of in [AbsTopIII], [the most fundamental special case of which arises from the tautologi- Torelli maps cal family of one-dimensional complex tori parametrized by the upper half-plane!], where a “Hodge structure” corresponds precisely to the specification of a partic- ular holomorphic structure in a situation in which one considers variations of the holomorphic structure on a fixed underlying real analytic structure. That is to say, later, in [IUTchIII], we shall see that the position occupied by a “Hodge theater” within a much larger framework that will be referred to as the [cf. “log-theta-lattice” the discussion of § I4 below] corresponds precisely to the specification of a partic- ular arithmetic holomorphic structure in a situation in which such arithmetic holomorphic structures are subject to deformation .

21 ̈ INTER-UNIVERSAL TEICHM 21 ULLER THEORY I I3. Basepoints and Inter-universality § I2, the present series of papers is concerned with considering As discussed in § “deformations of the arithmetic holomorphic structure” of a number field — i.e., so to speak, with performing . At a more concrete “surgery on the number field” level, this means that one must consider situations in which two distinct “theaters” ± ell NF-Hodge theaters — i.e., two distinct Θ for conventional ring/scheme theory ,or “filter” ,that — are related to one another by means of a “correspondence” fails ring structures . In the discussion so far of to be compatible with the respective the portion of the theory developed in the present paper, the main example of such -link a “filter” is given by the Θ . As mentioned earlier, more elaborate versions of the Θ-link will be discussed in [IUTchII], [IUTchIII]. The other main example of such a non-ring/scheme-theoretic “filter” in the present series of papers is the log -link , which we shall discuss in [IUTchIII] [cf. also the theory of [AbsTopIII]]. One important aspect of such non-ring/scheme-theoretic filters is the property incompatible with various constructions that depend on the that they are ring structure of the theaters that constitute the domain and codomain of such a filter. From the point of view of the present series of papers, perhaps the most impor- ́ tant example of such a construction is given by the various etale fundamental groups Galois groups — that appear in these theaters. Indeed, these — e.g., automorphism groups groups are defined, essentially, as of some separably closed field , i.e., the field that arises in the definition of the fiber functor associated to the basepoint determined by a geometric point that is used to define the ́ etale fun- damental group — cf. the discussion of [IUTchII], Remark 3.6.3, (i); [IUTchIII], Remark 1.2.4, (i); [AbsTopIII], Remark 3.7.7, (i). In particular, unlike the case etale with ring homomorphisms or morphisms of schemes with respect to which the ́ properties, in the case of non- fundamental group satisfies well-known functoriality “type of mathematical object” ring/scheme-theoretic filters, the only that makes sense simultaneously in both the domain and codomain theaters of the filter is the topological group . In particular, the only data that can be considered in notion of a etale fundamental groups on either side of a filter is the relating ́ etale-like struc- ́ constituted by the underlying abstract topological group ture associated to such an ́ etale fundamental group, i.e., devoid of any auxiliary data arising from the construction of the group “as an ́ etale fundamental group associated to a base- point determined by a geometric point of a scheme” . It is this fundamental aspect of the theory of the present series of papers — i.e., of relating the distinct set-theoretic universes associated to the distinct fiber functors/basepoints on either side of such a non-ring/scheme-theoretic filter — that we refer to as inter-universal . This inter-universal aspect of the theory manifestly leads to the issue of considering the extent to which one can understand various ring/scheme structures by considering only the underlying abstract topological group of some ́ etale fundamental group arising from such a ring/scheme structure — i.e., in other words, of considering the absolute anabelian geometry [cf. the Introductions to [AbsTopI], [AbsTopII], [AbsTopIII]] of the rings/schemes under consideration.

22 22 SHINICHI MOCHIZUKI At this point, the careful reader will note that the above discussion of the inter-universal aspects of the theory of the present series of papers depends, in “types of mathematical an essential way, on the issue of distinguishing different notion of a “type of mathematical object” objects” . and hence, in particular, on the This notion may be formalized via the language of , which we develop “species” in the final portion of [IUTchIV]. Another important phenomenon in the present series of pa- “inter-universal” pers — i.e., phenomenon which, like the absolute anabelian aspects discussed above, basepoints ” — is the phe- arises from a “deep sensitivity to particular choices of , i.e., of synchronization between conju- nomenon of conjugate synchronization gacy indeterminacies of distinct copies of various local Galois groups, which, as was § I1, will play an important role in the theory of [IUTchII], [IUTchIII]. mentioned in The various of the ́ etale theta function established in [EtTh] rigidity properties constitute yet another inter-universal phenomenon that will play an important role in theory of [IUTchII], [IUTchIII]. § p -adic Teichm ̈ uller Theory I4. Relation to Complex and In order to understand the sense in which the theory of the present series of papers may be thought of as a sort of uller theory” of number fields “Teichm ̈ equipped with an elliptic curve, it is useful to recall certain basic, well-known facts concerning the classical complex Teichm ̈ uller theory of Riemann surfaces of § 8]. Although such a Riemann surface is finite type [cf., e.g., [Lehto], Chapter V, from a complex, holomorphic point of view, this single complex one-dimensional two . dimension may be thought of consisting of underlying real analytic dimensions z Relative to a suitable canonical holomorphic coordinate x + iy on the Riemann = surface, the Teichm ̈ uller deformation may be written in the form z  ζ = ξ + iη = Kx + iy →

23 ̈ INTER-UNIVERSAL TEICHM 23 ULLER THEORY I two cohomological dimensions of the · of F . absolute Galois group G F nonar- of a G A similar statement holds in the case of the absolute Galois group k . In the case of k [i.e., k chimedean local field complex archimedean fields topological fields isomorphic to the field of complex numbers equipped with its may also be thought of as usual topology], the two combinatorial dimensions of k corresponding to the two topological/real dimensions of k . underlying · Alternatively, in both the nonarchimedean and archimedean cases, one may think as corresponding to the of the two underlying combinatorial dimensions of k × × × O value group k k / of and . · O group of units k k Indeed, in the nonarchimedean case, local class field theory implies that this last point of view is consistent with the interpretation of the two underlying combi- natorial dimensions via cohomological dimension; in the archimedean case, the consistency of this last point of view with the interpretation of the two underly- ing combinatorial dimensions via topological/real dimension is immediate from the definitions. This last interpretation in terms of groups of units and value groups is of particular relevance in the context of the theory of the present series of papers. That is to say, one may think of the Θ -link ∼  †  ‡ F → F tht mod ‡ † }  → { Θ q bad v ∈ V v v — which, as discussed in § I1, induces a full poly-isomorphism ∼ × × † ‡ F F → mod mod ∼ × × {O →O } bad ∈ V v F F v v “ Teichm ̈ uller deformation relative to a Θ -dilation ” —asasortof , i.e., a de- formation of the of the number field equipped with an elliptic ring structure initial curve constituted by the given -data in which one dilates the underlying Θ combinatorial dimension corresponding to the local value groups relative to a “ Θ - factor” , up to isomorphism, the underlying combinatorial di- fixed , while one leaves mension corresponding to the local groups of units [cf. Remark 3.9.3]. This point § I1 of “disentangling/dismantling” of view is reminiscent of the discussion in . of various structures associated to a number field ell ± NF-Hodge of Θ In [IUTchIII], we shall consider two-dimensional diagrams theaters which we shall refer to as log-theta-lattices . The two dimensions of such diagrams correspond precisely to the two underlying combinatorial dimensions of a ring . Of these two dimensions, the “theta dimension” consists of the Frobenius- picture associated to [more elaborate versions of] the Θ -link .Manyoftheimpor- tant properties that involve this “theta dimension” are consequences of the theory of [FrdI], [FrdII], [EtTh]. On the other hand, the “log dimension” consists of iter- ated copies of the log -link , i.e., diagrams of the sort that are studied in [AbsTopIII].

24 24 SHINICHI MOCHIZUKI “deformations of the That is to say, whereas the “theta dimension” corresponds to arithmetic holomorphic structure” of the given number field equipped with an el- liptic curve, this “log dimension” corresponds to “rotations of the two underlying of a ring that leave the arithmetic holomorphic struc- combinatorial dimensions” “juggling of ” ,  induced by log ture in  fixed — cf. the discussion of the ultimate conclusion of the theory of [IUTchIII] is that I3. The § [AbsTopIII], the “a priori unbounded deformations” of the arithmetic holomorphic canonical bounds structure given by the Θ-link in fact admit ,which of the “hyperbolicity” may be thought of as a sort of reflection of the given number field equipped with an elliptic curve — cf. [IUTchIII], Corollary 3.12. Such canonical bounds may be thought of as analogues for a number field of canonical bounds that arise from differentiating Frobenius liftings -adic hyperbolic curves — cf. the discus- in the context of p § I5. Moreover, such canonical bounds are sion in the final portion of [AbsTopIII], obtained in [IUTchIII] as a consequence of varying arithmetic holomorphic struc- the explicit description of a within a ture fixed mono-analytic “container” § I2! — furnished by [IUTchIII], Theorem 3.11 [cf. also — cf. the discussion of the discussion of [IUTchIII], Remarks 3.12.2, 3.12.3, 3.12.4], i.e., a situation that entirely formally analogous uller theory given to the summary of complex Teichm ̈ is above. The significance of the log-theta-lattice is best understood in the context of uller theory developedinthe inter-universal Teichm ̈ the analogy between the p -adic Teichm ̈ uller theory of [ p Ord], [ p present series of papers and the Teich]. Here, we recall for the convenience of the reader that the -adic Teichm ̈ uller theory p p p Teich] may be summarized, [very!] roughly speaking, as a sort of of [ Ord], [ , to the case of “quite general” p generalization ,of -adic hyperbolic curves the classical p -adic theory surrounding the canonical representation 1 Z (( P , \{ 0 , 1 ) ∞} ) ( PGL ) → π → (( M ) ) π Q 2 Q 1 p 1 ell p p —wherethe“ π − )’s” denote the ́ etale fundamental group , relative to a suitable ( 1 ) denotes the moduli stack of elliptic curves over Q ; the first M basepoint; ( Q p ell p arrow denotes the morphism induced by the elliptic curve over the projective line Legendre form of the Weierstrass minus three points determined by the classical equation; the second arrow is the representation determined by the p -power torsion . In particular, the reader who ) over ( M tautological elliptic curve of the points ell Q p is familiar with the theory of the classical representation of the above display, but p p Teich], may nevertheless appreciate, to a substantial not with the theory of [ Ord], [ degree, the analogy between the inter-universal Teichm ̈ uller theory developed in the present series of papers and the -adic Teichm ̈ uller theory of [ p Ord], [ p Teich] by p thinking in terms of the well-known classical properties of this classical representation . In some sense, the between the “quite general” p -adic hyperbolic curves that gap 1 , may ) ∞} , \{ 0 1 -adic Teichm ̈ appear in P uller theory and the classical case of ( p Q p

25 ̈ INTER-UNIVERSAL TEICHM 25 ULLER THEORY I be thought of, roughly speaking, as corresponding, relative to the analogy with the theory of the present series of papers, to the gap between arbitrary number fields Q rational number field . This point of view is especially interesting in and the the context of the discussion of I5 below. § The analogy between the uller theory developed in inter-universal Teichm ̈ the present series of papers and the uller theory of [ p Ord], [ p Teich] p -adic Teichm ̈ § I5, i.e., where is described to a substantial degree in the discussion of [AbsTopIII], “future Teichm ̈ uller-like extension of the mono-anabelian theory” the may be un- derstood as referring precisely to the inter-universal Teichm ̈ uller theory developed in the present series of papers. The starting point of this analogy is the correspon- dence between a number field equipped with a [once-punctured] elliptic curve [in the hyperbolic curve over a positive characteristic perfect present series of papers] and a field equipped with a nilpotent ordinary indigenous bundle p -adic Teichm ̈ uller [in number field theory] — cf. Fig. I4.1 below. That is to say, in this analogy, the — which may be regarded as being equipped with a finite collection of “exceptional” bad — corre- , namely, in the notation of § valuations V I1, the valuations lying over mod sponds to the hyperbolic curve over a positive characteristic perfect field —which may be thought of as a one-dimensional function field over a positive characteristic finite collection of “exceptional” valuations ,namely, perfect field, equipped with a the valuations corresponding to the cusps of the curve. [once-punctured] elliptic curve in the present series On the other hand, the of papers corresponds to the in p -adic Te- nilpotent ordinary indigenous bundle uller theory. Here, we recall that an indigenous bundle may be thought of as a ichm ̈ “virtual analogue” sort of of the first cohomology group of the tautological elliptic curve over the moduli stack of elliptic curves. Indeed, the canonical indigenous bundle over the moduli stack of elliptic curves arises precisely as the first de Rham cohomology module of this tautological elliptic curve. Put another way, from the point of view of , an indigenous bundle may be thought of as fundamental groups a sort of “virtual analogue” of the abelianized fundamental group of the tau- tological elliptic curve over the moduli stack of elliptic curves. By contrast, in the present series of papers, it is of crucial importance to use the entire nonabelian etale fundamental group profinite ́ — i.e., not just its abelizanization! — of the given once-punctured elliptic curve over a number field. Indeed, only by working etale fundamental group can one avail oneself of the crucial with the entire profinite ́ developed in [EtTh], [AbsTopIII] [cf. the discussion absolute anabelian theory of § I3]. This state of affairs prompts the following question: To what extent can one extend the indigenous bundles that appear in clas- sical complex p -adic Teichm ̈ uller theory to objects that serve as “vir- and tual analogues” of the entire nonabelian fundamental group of the tautological once-punctured elliptic curve over the moduli stack of [once- punctured] elliptic curves? Although this question lies beyond the scope of the present series of papers, it is

26 26 SHINICHI MOCHIZUKI the hope of the author that this question may be addressed in a future paper. Inter-universal Teichm ̈ -adic Teichm ̈ uller theory uller theory p C hyperbolic curve number field over a F positive characteristic perfect field [once-punctured] nilpotent ordinary indigenous bundle elliptic curve over F P over C X -link arrows of the mixed characteristic extension Θ structure of a ring of log-theta-lattice Witt vectors -link the Frobenius morphism log arrows of the log-theta-lattice in positive characteristic the resulting canonical lifting the entire canonical Frobenius action ; + log-theta-lattice canonical Frobenius lifting over the ordinary locus relatively straightforward relatively straightforward original construction of original construction of log-theta-lattice canonical liftings highly nontrivial highly nontrivial absolute anabelian alien arithmetic description of reconstruction of holomorphic structure via absolute anabelian geometry canonical liftings Fig. I4.1: Correspondence between inter-universal Teichm ̈ uller theory and p -adic Teichm ̈ uller theory Now let us return to our discussion of the log-theta-lattice, which, as discussed above, consists of two types of arrows, namely, Θ -link arrows and log -link ar- rows. As discussed in [IUTchIII], Remark 1.4.1, (iii) — cf. also Fig. I4.1 above, as well as Remark 3.9.3, (i), of the present paper — the Θ-link arrows correspond n n 1 n +1 − n /p p Z /p Z to Z Z ” , i.e., the mixed characteris- to the “transition from p tic extension structure of a ring of Witt vectors , while the log -link arrows, i.e.,

27 ̈ INTER-UNIVERSAL TEICHM 27 ULLER THEORY I the portion of theory that is developed in detail in [AbsTopIII], and which will be incorporated into the theory of the present series of papers in [IUTchIII], corre- in positive characteristic . As we shall see in spond to the Frobenius morphism [cf. [IUTchIII], Remark 1.4.1, fail to commute [IUTchIII], these two types of arrows “intertwining” , of the Θ-link and log -link arrows (i)]. This noncommutativity, or of the log-theta-lattice may be thought of as the analogue, in the context of the theory of the present series of papers, of the well-known “intertwining between the mixed characteristic extension structure of a ring of Witt vectors and the Frobenius morphism in positive characteristic” that appears in the classical -adic theory. In p particular, taken as a whole, the log-theta-lattice in the theory of the present series of papers may be thought of as an analogue, for number fields equipped with a [once-punctured] elliptic curve, of the canonical lifting , equipped with a canon- — hence also the canonical Frobenius lifting over the ical Frobenius action ordinary locus of the curve — associated to a positive characteristic hyperbolic curve equipped with a nilpotent ordinary indigenous bundle in -adic Teichm ̈ uller p theory [cf. Fig. I4.1 above; the discussion of [IUTchIII], Remarks 3.12.3, 3.12.4]. Finally, we observe that it is of particular interest in the context of the present § 3, that yields an absolute discussion that a theory is developed in [CanLift], for the canonical liftings of p -adic Teichm ̈ uller the- anabelian reconstruction . That is to say, whereas the original construction of such canonical liftings ory p Ord], given in [ 3, is relatively straightforward ,the anabelian reconstruction given § 2 of the logarith- § in [CanLift], 3, of, for instance, the canonical lifting modulo p highly nontrivial anabelian argument . This state of mic special fiber consists of a affairs is strongly reminiscent of the stark contrast between the relatively straight- forward construction of the log-theta-lattice given in the present series of papers and the description of an “alien arithmetic holomorphic structure” given in [IUTchIII], Theorem 3.11 [cf. the discussion in the earlier portion of the present § I4], which highly nontrivial results in absolute anabelian geometry — is achieved by applying cf. Fig. I4.1 above. In this context, we observe that the absolute anabelian theory § of [AbsTopIII], 1, which plays a central role in the theory surrounding [IUTchIII], § 3, to the absolute anabelian Theorem 3.11, corresponds, in the theory of [CanLift], given in [AbsAnab], § 2 [i.e., in essence, reconstruction of the logarithmic special fiber the theory of absolute anabelian geometry over finite fields developed in [Tama1]; cf. also [Cusp], § 2]. Moreover, just as the absolute anabelian theory of [AbsTopIII], § 1, follows essentially by combining a version of “Uchida’s Lemma” [cf. [AbsTopIII], Belyi cuspidalization Proposition 1.3] with the theory of — i.e., § 1 = Uchida Lem. + Belyi cuspidalization [AbsTopIII], — the absolute anabelian geometry over finite fields of [Tama1], [Cusp], follows “Uchida’s Lemma” with an application [to essentially by combining a version of the counting of rational points] of the Lefschetz trace formula for [powers of] the Frobenius morphism on a curve over a finite field — i.e., [Tama1], [Cusp] = Uchida Lem. + Lefschetz trace formula for Frob. § I5. That is to say, it is perhaps worthy of — cf. the discussion of [AbsTopIII], note that in the analogy between the inter-universal Teichm ̈ uller theory developed in the present series of papers and the p -adic Teichm ̈ uller theory of [ p Ord], [ p Teich], [CanLift], the application of the theory of Belyi cuspidalization over number fields

28 28 SHINICHI MOCHIZUKI and mixed characteristic local fields may be thought of as corresponding to the Lefschetz trace formula for [powers of] the Frobenius morphism on a curve over a finite field, i.e., Lefschetz trace formula for Frobenius Belyi cuspidalization ←→ [Here, we note in passing that this correspondence may be related to the corre- spondence discussed in [AbsTopIII], § I5, between Belyi cuspidalization and the Verschiebung on positive characteristic indigenous bundles by considering the ge- ometry of Hecke correspondences modulo p , i.e., in essence, graphs of the Frobenius p !] It is the hope of the author that these analogies and morphism in characteristic correspondences might serve to stimulate further developments in the theory. § I5. Other Galois-theoretic Approaches to Diophantine Geometry The notion of dates back to a famous “letter to Falt- anabelian geometry ings” [cf. [Groth]], written by Grothendieck in response to Faltings’ work on the [cf. [Falt]]. Anabelian geometry was apparently originally con- Mordell Conjecture diophantine ceived by Grothendieck as a new approach to obtaining results in geometry such as the Mordell Conjecture. At the time of writing, the author is not aware of any expositions by Grothendieck that expose this approach in detail. Nevertheless, it appears that the thrust of this approach revolves around applying Section Conjecture for hyperbolic curves over number fields to obtain a con- the tradiction by applying this Section Conjecture to the “limit section” of the Galois sections associated to any infinite sequence of rational points of a proper hyperbolic curve over a number field [cf. [MNT], § 4.1(B), for more details]. On the other hand, to the knowledge of the author, at least at the time of writing, it does not appear that any rigorous argument has been obtained either by Grothendieck or by other mathematicians for deriving a new proof of the Mordell Conjecture from the [as yet unproven] Section Conjecture for hyperbolic curves over number fields. Nev- ertheless, one result that has been obtained is a new proof by M. Kim [cf. [Kim]] Siegel’s theorem concerning Q of -rational points of the projective line minus three points — a proof which proceeds by obtaining certain bounds on the cardinality of the set of Galois sections, without applying the Section Conjecture or any other results from anabelian geometry. In light of the historical background just discussed, the theory exposed in the present series of papers — which yields, in particular, a method for applying results in absolute anabelian geometry to obtain diophantine results such as those given in [IUTchIV] — occupies a , relative to somewhat curious position the historical development of the mathematical ideas involved. That is to say, at a purely formal level, the implication anabelian geometry = ⇒ diophantine results at first glance looks something like a “confirmation” of Grothendieck’s original intuition. On the other hand, closer inspection reveals that the approach of the theory of the present series of papers — that is to say, the precise content of the relationship between anabelian geometry and diophantine geometry established in

29 ̈ INTER-UNIVERSAL TEICHM 29 ULLER THEORY I differs quite fundamentally from the sort of approach the present series of papers — that was apparently envisioned by Grothendieck. Perhaps the most characteristic aspect of this difference lies in the central role played by p -adic fields in the present series of papers. anabelian geometry over That is to say, unlike the case with number fields, one central feature of anabelian -adic fields is the fundamental gap between relative and absolute geometry over p results [cf., e.g., [AbsTopI], Introduction]. This fundamental gap is closely related “arithmetic Teichm ̈ [cf. to the notion of an uller theory for number fields” I4 of the present paper; [AbsTopIII], § I3, § I5] — i.e., a theory of the discussion of § not for the “arithmetic holomorphic structure” of a hyperbolic curve deformations over a number field, but rather for the “arithmetic holomorphic structure” of the number field itself ! To the knowledge of the author, there does not exist any mention p -adic anabelian geometry; the notion of an of such ideas [i.e., relative vs. absolute arithmetic Teichm ̈ uller theory for number fields] in the works of Grothendieck. As discussed in I4, one fundamental theme of the theory of the present series § of papers is the issue of the explicit description of the relationship between the additive structure and multiplicative structure of a ring/number field/local field . the Relative to the above discussion of the relationship between anabelian geometry and diophantine geometry, it is of interest to note that this issue of understand- ing/describing the relationship between and multiplication is, on the one addition proofs of various results in hand, a central theme in the [cf., anabelian geometry e.g., [Tama1], [ p GC], [AbsTopIII]] and, on the other hand, a central aspect of the diophantine results obtained in [IUTchIV]. From a historical point of view, it is also of interest to note that results from ab- solute anabelian geometry are applied in the present series of papers in the context of the of the Frobenius-picture that arise by considering the canonical splittings etale-picture [cf. the discussion in I1 preceding Theorem A]. This state of affairs ́ § is reminiscent — relative to the point of view that the Grothendieck Conjecture “anabelian version” of the constitutes a sort of for abelian varieties Tate Conjecture [cf. the discussion of [MNT], § 1.2] — of the role played by the Tate Conjecture for diophantine results of [Falt], namely, by means abelian varieties in obtaining the semi-simplicity of the various properties of the Tate module that arise as formal consequences of the Tate Conjecture. That is to say, such semi-simplicity proper- ties may also be thought of as that arise from Galois-theoretic “canonical splittings” considerations [cf. the discussion of “canonical splittings” in the final portion of [CombCusp], Introduction]. Certain aspects of the relationship between the inter-universal Teichm ̈ uller theory of the present series of papers and other Galois-theoretic approaches to dio- phantine geometry are best understood in the context of the analogy , discussed in § I4, between inter-universal Teichm ̈ uller theory and p -adic Teichm ̈ uller theory . One way to think of the starting point of p ullerisasanattemptto -adic Teichm ̈ ( Z )onthe upper p -adic analogue of the theory of the action of construct a SL 2 half-plane , i.e., of the natural embedding ) ) : SL R ( Z ( ↪ → SL ρ 2 R 2

30 30 SHINICHI MOCHIZUKI Z ) as a discrete subgroup. This leads naturally to consideration of the SL of ( 2 representation ∏ ∏ ∧ ̂ ( : = ( Z ) SL → SL ( ) Z )= ρ Z SL ρ Z p 2 2 2 p ̂ Z p Primes ∈ p ∧ SL —wherewewrite ( for the profinite completion of SL ) ( Z ). If one thinks Z 2 2 ∧ Z ) ( geometric ́ etale fundamental group of the moduli stack of elliptic as the SL of 2 over a field of characteristic zero, then the p -adic Teichm ̈ uller theory of curves - to more general p Teich] does indeed constitute a generalization of ρ [ p Ord], [ p Z p adic hyperbolic curves. representation-theoretic point of view, the next natural direction From a in which to further develop the theory of [ p Ord], [ p Teich] consists of attempting to Ord], [ ( Z Teich] ) obtained in [ p p SL generalize the theory of representations into p 2 n Z )for arbitrary ( ≥ 2. This is SL to a theory concerning representations into n p precisely the motivation that lies, for instance, behind the work of Joshi and Pauly [cf. [JP]]. , the rep- original motivating representation On the other hand, unlike the ρ R Congruence , i.e., put another way, the so-called is far from injective resentation ρ ̂ Z . This failure of injectivity means Subgroup Property fails to hold in the case of SL 2 that working with ∧ SL only allows one to access a relatively limited portion of . ) ( Z ρ 2 ̂ Z From this point of view, a more natural direction in which to further develop the theory of [ Ord], [ p Teich] is to consider the “anabelian version” p ∧ ( Z ) → Out(Δ SL ) : ρ , Δ 2 1 1 — i.e., the natural outer representation on the geometric ́ etale fundamen- ρ of ̂ Z once-punctured elliptic curves of the tautological family of over the Δ tal group , 1 1 moduli stack of elliptic curves over a field of characteristic zero. Indeed, unlike the is , one knows [cf. [Asada]] that ρ “arithmetic .Thus,the injective case with ρ Δ ̂ Z uller theory Teichm ̈ number fields equipped with a [once-punctured] el- for liptic curve” constituted by the inter-universal Teichm ̈ uller theory developed in the present series of papers may [cf. the discussion of § I4!] be regarded as a realization of this sort of “anabelian” approach to further developing the p -adic Teichm ̈ uller theory of [ p Ord], [ p Teich]. In the context of these two distinct possible directions for the further develop- -adic Teichm ̈ uller theory of [ p Ord], [ p Teich], it is of interest to recall p ment of the the following elementary fact: G is a free pro- p group of rank ≥ 2, then a [continuous] representation If : G → GL ( Q ) ρ G n p can never be injective ! Indeed, assume that ρ ) is injective and write ... ⊆ H Q ⊆ ... ⊆ Im( ρ ( ) ⊆ GL n G j p G .Thensince for an exhaustive sequence of open normal subgroups of the image of ρ G

31 ̈ INTER-UNIVERSAL TEICHM 31 ULLER THEORY I H the GL are closed subgroups of ( Q -adic Lie groups, it follows that the ), hence p p j n ab Q H ⊗ of the abelianization -dimension Q dim( ) of the tensor product with Q p p p j may be computed at the level of , hence is bounded by the Q - Lie algebras H of j p ab 2 GL -adic Lie group p dimension of the ,in ), i.e., we have dim( H ( Q ⊗ Q n ) ≤ p p n j ∼ contradiction to the well-known fact since G Im( ρ 2, it )is free pro- p of rank ≥ = G ab j . Note, moreover, that ⊗ Q →∞ ) →∞ as holds that dim( H p j asymptotic behavior of the this sort of argument, i.e., concerning the — or, more generally, of the Lie algebras associated to abelianizations the pro-algebraic groups determined by associated Tannakian categories of representations — of open subgroups ,is characteristic of the sort of proofs that typically occur in anabelian geometry [cf., e.g., the proofs p of [Tama1], [ 3!]. GC], [CombGC], as well as [Cusp], § can never be injective is a ρ That is to say, the above argument to the effect that G typical instance of the more general phenomenon that into GL so long as one restricts oneself to representation theory Q ) ( p n -valued points Q [or even more general groups that arise as groups of p asymptotic of pro-algebraic groups], one can never access the sort of that form the “technical core” [cf., e.g., the proofs of phenomena p GC], [CombGC], as well as [Cusp], § 3!] of various important [Tama1], [ results in anabelian geometry. Put another way, the two “directions” discussed above — i.e., representation- theoretic and anabelian —appeartobe essentially mutually alien to one another. In this context, it is of interest to observe that the diophantine results de- rived in [IUTchIV] from the inter-universal Teichm ̈ uller theory developed in the present series of papers concern essentially asymptotic behavior , i.e., they do not concern properties of “a specific rational point over a specific number field”, but rather properties of the asymptotic behavior of “varying rational points over varying number fields” . One important aspect of this asymptotic nature of the dio- no distinguished number phantine results derived in [IUTchIV] is that there are that occur in the theory, i.e., the theory — being in essentially asymptotic fields nature! — is “invariant” with respect to the operation of passing to finite exten- absolute sions of the number field involved [which, from the point of view of the smaller of Q , corresponds precisely to the operation of passing to Galois group G Q , as in the above discussion!]. This contrasts sharply and smaller open subgroups with the “representation-theoretic approach to diophantine geometry” constituted by such works as [Wiles], where specific rational points over the specific number field Q — or, for instance, in generalizations of [Wiles] involving Shimura varieties, over specific number fields characteristically associated to the Shimura varieties involved — play a central role. Acknowledgements: The research discussed in the present paper profited enormously from the gen- erous support that the author received from the Research Institute for Mathematical

32 32 SHINICHI MOCHIZUKI , a Joint Usage/Research Center located in Kyoto University. At a per- Sciences Fumiharu Kato , Akio Tamagawa , Go Yamashita , sonal level, I would like to thank ıdi , Mohamed Sa ̈ , Ivan Fesenko , Fucheng Tan , Emmanuel Lepage , Yuichiro Hoshi and Arata Minamide for many stimulating discussions concerning the material pre- sented in this paper. In particular, I would like to thank Emmanuel Lepage for his stimulating comments [summarized in Remark 2.5.3] on [SemiAnbd]. Also, I feel Go Yamashita , Mohamed Sa ̈ ıdi ,and Yuichiro Hoshi for their deeply indebted to meticulous reading of and numerous comments concerning the present paper. In particular, the introduction of the theory of κ -coric functions was motivated by var- ious stimulating discussions with Yuichiro Hoshi . Finally, I would like to express my deep gratitude to Ivan Fesenko for his quite substantial efforts to disseminate — for instance, in the form of a survey that he wrote — the theory discussed in the present series of papers.

33 ̈ INTER-UNIVERSAL TEICHM 33 ULLER THEORY I Section 0: Notations and Conventions Monoids and Categories: and categories We shall use the notation and terminology concerning monoids 0. of [FrdI], § P We shall refer to a topological space equipped with a continuous map P P ⊇ S → P × M [whose topological pseudo-monoid as a if there exists a topological abelian group ] and an embedding of topological multiplicatively group operation will be written : P↪ → M such that S = spaces ( a, b ) ∈ P × P | ι ( a ) · ι ( b ) ∈ ι ( P ) ⊆ M } ,and ι { S → P is obtained by restricting the group operation M × M the map M on → M P by means of ι . Here, if M is equipped with the discrete topology ,then to P simply as a . In particular, every we shall refer to the resulting pseudo-monoid topological pseudo-monoid determines, in an evident fashion, an underlying pseudo- P pseudo-monoid . Then we shall say that the pseudo-monoid monoid. Let be a ι may be taken such that for each positive integer n ,every is divisible if P and M M admits an n -th root in M ,and,moreover,anelement a element of M lies ∈ n P ). We shall say that the pseudo-monoid P ( is lies in ι a in P ( ι ) if and only if of torsion ⊆ M μ may be taken such that the subgroup M if cyclotomic ι and M ). P ( ι ⊆ ) P ( ι ⊆ ι ( P ), and μ · Z / Q is isomorphic to the group M elements of μ , M M of the isomorph We shall refer to an isomorphic copy of some object as an object. and D are categories , then we shall refer to as an isomorphism C→D any If C C→D . [Note that this termniology isomorphism class of equivalences of categories natural in the differs from the standard terminology of category theory, but will be .] Thus, from the point of view context of the theory of the present series of papers 2 of 1 -categories” [cf. [FrdI], Appendix, Definition “coarsifications of -categories of C→D ” is precisely an “isomorphism in the usual sense” A.1, (ii)], an “isomorphism of the [1-]category constituted by the coarsification of the 2-category of all small 1-categories relative to a suitable universe with respect to which and D are small. C C be a category ; A, B ∈ Ob( C ). Then we define a poly-morphism A → B Let A B [i.e., a subset of the set of morphisms → to be a collection of morphisms A → B ]; if all of the morphisms in the collection are isomorphisms, then we shall poly-isomorphism ;if A = B , then we shall re- refer to the poly-morphism as a ∼ . We define the full → B as a poly-automorphism A fer to a poly-isomorphism ∼ → B to be the poly-morphism given by the collection of all poly-isomorphism A ∼ → B . The composite of a poly-morphism { f with a : A → B } A isomorphisms i I i ∈ is defined to be the poly-morphism given by the B → C } : { g poly-morphism ∈ j j J C → A . } f : ◦ g { [i.e., where “multiplicities” are ignored] set j i i,j ) I × J ( ∈ C Let category . We define a capsule of objects of C to be a finite collection be a { A denotes the } | J | [i.e., where J is a finite index set] of objects A ;if of C j J ∈ j j

34 34 SHINICHI MOCHIZUKI J cardinality of as a | J | -capsule ; , then we shall refer to a capsule with index set J def = A C } morphism of capsules of objects of .A J ) ( { also, we shall write π J ∈ j j 0 ′ ′ ′ } } →{ A { A ′ j j j ∈ J J ∈ j ′ j → J ι , together with, for each is defined to consist of an injection ∈ J ,a J↪ : ′ C → A form a C of objects of . Thus, the capsules of objects of A morphism j j ( ι ) Capsule( category capsule-full poly-morphism C ). A ′ ′ ′ A { } →{ A } ′ j j ∈ ∈ J J j j C ) is defined to be the poly-morphism associated between two objects of Capsule( ′ which consists of the set of morphisms of ι J↪ → J to some [fixed] : injection ∼ ′ C [arbitrary] isomorphisms A ) given by collections of Capsule( A ∈ j ,for → j ) ι j ( .A is a capsule-full poly-morphism for which the J capsule-full poly-isomorphism bijection . associated injection between index sets is a X is a connected noetherian algebraic stack which is generically scheme-like If [cf. [Cusp], § 0], then we shall write B ( ) X category of finite ́ for the X [and morphisms over X ]; if A is a etale coverings of def B = (Spec( A )). , then we shall write ) B A noetherian [commutative] ring [with unity] ( 0 ( ⊆B X )may ) § 0] the subcategory of connected objects Thus, [cf. [FrdI], ( X B connected finite ́ of X [and be thought of as the subcategory of etale coverings morphisms over X ]. topological group . Then let us write Let Π be a temp (Π) B [i.e., of cardinality whose objects are for the category ≤ the cardinality countable of the set of natural numbers], discrete sets equipped with a continuous Π-action, and whose morphisms are morphisms of Π-sets [cf. [SemiAnbd], § 3]. If Π may be written as an inverse limit of an inverse system of surjections of countable discrete tempered topological groups, then we shall say that Π is [cf. [SemiAnbd], Definition temp (Π), where Π is a equivalent to a category of the form C B 3.1, (i)]. A category [cf. [SemiAnbd], Defi- tempered topological group, is called a connected temperoid is a connected temperoid, then C is naturally equivalent C nition 3.1, (ii)]. Thus, if 0 ) [cf. [FrdI], § 0]. Moreover, if Π is Galois-countable [cf. Remark 2.5.3, (i), C to ( reconstruct [cf. Remark 2.5.3, (i), (T5)] the topological group Π , (T1)], then one can temp temp 0 (Π) or B (Π) [i.e., up to inner automorphism, category-theoretically from B temp (Π)]; in particular, for any Galois- the subcategory of connected objects of B countable C , it makes sense to [cf. Remark 2.5.3, (i), (T1)] connected temperoid write 0 ( ) C ) ,π C ( π 1 1 for the topological groups , up to inner automorphism, obtained by applying this reconstruction algorithm [cf. Remark 2.5.3, (i), (T5)].

35 ̈ INTER-UNIVERSAL TEICHM 35 ULLER THEORY I C are connected temperoids , then it is natural to define , In this context, if C 1 2 a morphism C →C 1 2 →C C to be an isomorphism class of functors that preserves finite limits and 1 2 countable colimits. [Note that this differs — but only slightly! — from the definition given in [SemiAnbd], Definition 3.1, (iii). This difference does not, however, have any effect on the applicability of results of [SemiAnbd] in the context of the present series of papers.] In a similar vein, we define a morphism 0 0 →C C 1 2 0 0 [where we recall that we have natural equivalences → C ) to be a morphism ( ) C ( 1 2 ∼ 0 of categories C C “isomor- → ) ( for i =1 , 2]. One verifies immediately that an i i relative to this terminology is equivalent to an “isomorphism of categories” phism” “Monoids and in the sense defined at the beginning of the present discussion of ,Π [cf. Remark 2.5.3, (i), (T1)] are Galois-countable . Finally, if Π Categories” 1 2 tempered topological groups, then we recall that there is a natural bijective corre- spondence between → Π , (a) the set of continuous outer homomorphisms Π 2 1 temp temp (Π (Π ) →B ), and (b) the set of morphisms B 1 2 temp 0 temp 0 →B (Π (Π ) ) (c) the set of morphisms B 2 1 — cf. Remark 2.5.3, (ii), (E7); [SemiAnbd], Proposition 3.2. ′ categories are C . Then we shall say that two and =1 Suppose that for 2, i , C i i ′ ′ ′ →C abstractly equivalent →C C , φ are : : φ isomorphism classes of functors C 2 1 2 1 ∼ ′ ′ φ : C ◦ →C .We α such that = φ ◦ α 2, there exist isomorphisms i if, for =1 α , i 1 2 i i shall also apply this terminology to morphisms between [connected] temperoids, as well as to morphisms between subcategories of connected objects of [connected] temperoids. Numbers: We shall use the abbreviations NF (“number field”), MLF (“mixed-characteris- tic [nonarchimedean] local field”), CAF (“complex archimedean field”), as defined § 0; [AbsTopIII], § 0. We shall denote the set of prime numbers by in [AbsTopI], . Primes F be a number field Let [i.e., a finite extension of the field of rational numbers]. Then we shall write ⋃ non arc ( V ) F ) ( )= F ( V F V valuations F , that is to say, the union of the sets of archimedean for the set of of non arc ] valuations of V . Here, we note ]and nonarchimedean [i.e., F ( F ) ) ( V F [i.e., that this terminology “valuation”, as it is applied in the present series of papers, “place” or “absolute value” in the work of other corresponds to such terminology as and for the completion of F at v F authors. Let v ∈ V ( F ). Then we shall write v of norm 1 with respect ≤ if it is ] is integral [at v say that an element of or F F v to the valuation v , L is any [possibly infinite] Galois extension of F ;if,moreover,

36 36 SHINICHI MOCHIZUKI L for the completion of L at then, by a slight abuse of notation, we shall write v non , then we shall write p L )thatliesover v .If v ∈ V ( F ) ∈ some valuation V ( v arc F v .If v ∈ V ( for the ) of , then we shall write p F ∈ residue characteristic v v for the unique positive real element of F 1 whose natural logarithm is equal to v [i.e., “ =2 . 71828 ... e projective or inductive limits, ”]. By passing to appropriate ”isan F ” in situations where “ p ”, “ we shall also apply the notation “ F F )”, “ ( V v v infinite extension of Q . Curves: We shall use the terms hyperbolic curve , cusp , stable log curve ,and smooth log curve as they are defined in [SemiAnbd], § 0. We shall use the term hyperbolic orbicurve as it is defined in [Cusp], § 0.

37 ̈ INTER-UNIVERSAL TEICHM 37 ULLER THEORY I Section 1: Complements on Coverings of Punctured Elliptic Curves In the present § 1, we discuss certain routine complements — which will be of use in the present series of papers — to the theory of coverings of once-punctured 2. , as developed in [EtTh], elliptic curves § , 5bean X a hyperbolic curve of type (1 6; 1) over a Let l ≥ integer prime to -tors) (1 a hyperbolic orbicurve of type [cf. [EtTh], ,l ; C of characteric zero field k ± ,whose k C [cf. [CanLift], Remark 2.1.1; [EtTh], the k Definition 2.1] over -core determines, k discussion at the beginning of X .Thus, C § 2] also forms a -core of def = C × X of type (1 ,l -tors) [cf. X k up to hyperbolic orbicurve -isomorphism, a C for the absolute Galois k G . Moreover, if we write [EtTh], Definition 2.1] over k for the arithmetic k group of [relative to an appropriate choice of basepoint], Π ) − ( fundamental group of a geometrically connected, geometrically normal, generically k -algebraic stack of finite type “( − )” [i.e., the ́ etale fundamental group scheme-like − ))], and Δ (( )” [i.e., the kernel − for the geometric fundamental group of “( π 1 ) ( −  G natural cartesian diagrams ], then we obtain of the natural surjection Π k ( ) − X −→ Δ Δ −→ Π −→ X Π X X X X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       C Π Δ Π C −→ −→ Δ −→ C C C C etale coverings of hyperbolic orbicurves and open immersions of profinite of finite ́ groups. Finally, let us make the following assumption: ab Z /l Z on Δ ) [where the superscript “ab” ( ⊗ G ) The natural action of ( ∗ k X denotes the abelianization] is trivial . nonzero — i.e., a cusp that arises from a be a nonzero cusp of C  Next, let Q element ” that appears in the definition of a “hyperbolic orbicurve of the quotient “ 0  for the unique “zero ” given in [EtTh], Definition 2.1. Write ,l -tors) of type (1 ± ′ ′′ that lie over   ; for the two cusps of X ,  ; X [i.e., “non-nonzero cusp”] of cusp” and ab Z Δ ⊗ (  /l Z )  Δ Δ  X X ab ⊗ ( for the quotient of Δ /l Z ) by the images of the inertia groups of all nonzero Z X ′ ′′ natural exact sequence , of X . Thus, we obtain a   cusps = ′ ′′ −→ I 0 I 0 −→ × Δ −→ −→ Δ ) ⊗ ( Z /l Z E    ′ ′′ —wherewewrite E ,and I for for the genus one compactification of , I X   ′ ′′ [so we have ,   of the inertia groups of the cusps the respective images in Δ  ∼ ∼ ′′ ′ I Z Z /l ]. I isomorphisms noncanonical = =   ∼ Z act naturally on the above ) Z / 2 /C , Gal( )( X Next, let us observe that G = k exact sequence .Write ι ∈ Gal( X /C ) for the unique nontrivial element. Then ι ∼ ′ ′′ ′ ′′ I ; if we use this isomorphism to identify I , I , I induces an isomorphism =     ′ ′′ I ”oftheaboveexact × acts on the term “ I ι then one verifies immediately that   sequence by switching the two factors . Moreover, one verifies immediately that ι

38 38 SHINICHI MOCHIZUKI ⊗ ( Z /l Z ) via multiplication by − 1. In particular, since l is odd , it follows acts on Δ E decomposition into eigenspaces determines a ι that the action by on Δ  ∼ + − → Δ × Δ Δ    + − (respectively, Δ acts on Δ — i.e., where ) by multiplication by +1 (respectively, ι   1). Moreover, the natural composite maps − + + ′ ′′ Δ  Δ ; ↪ → I Δ Δ ↪ →  I       ∼ ∼ + + ′ ′′ I I Δ determine → Δ → , . Since the natural action of G isomorphisms k     clearly commutes with the action of ι , we thus conclude that the quotient on Δ  + Δ determines quotients  Δ  Δ  X  Π  J J ;Π  C X X C  G ,Π — where the surjections Π  G induce natural exact sequences k C X k + + G → J → → G J → 1, 1 → 1; we have a → Δ × Gal( X /C ) → Δ → 1 k X k C   J natural inclusion ↪ → J . C X Next, let us consider the cusp “2  C — i.e., the cusp whose inverse images in ”of ′ ′′ correspond to the points of  ,  by 2, relative to the E obtained by multiplying X 0 , ). Since 2  = ± 1(mod l ) X group law of the elliptic curve determined by the pair ( l ≥ 5], it follows that the decomposition group [a consequence of our assumption that ” determines a section  associated to this cusp “2 : σ G → J k C J G  of the natural surjection is only . Here, we note that although, a priori, σ k C up to composition with an inner automorphism of J determined determined by 2  C + by an element of Δ Gal( X /C ), in fact, since [in light of the assumption ( ∗ )!] ×  + G the natural [outer] action of × Gal( X /C )is trivial , we conclude that σ on Δ k  J determined by , and that the subgroup Im( σ ) ⊆ by 2  completely determined is C . Moreover, by considering the decomposition groups normal in J the image of σ is C lying over 2  , we conclude that Im( σ ) lies inside the associated to the cusps of X J ⊆ ⊆ J ) . Thus, the subgroups Im( σ ) ⊆ J /C ,Im( σ ) × Gal( X J subgroup X C C X determine [the horizontal arrows in] cartesian diagrams X −→ X Δ Δ Π −→ −→ Π X X X X − → → − − → ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       −→ −→ Π Δ Π C −→ Δ C C C C C → − → − → − of finite ́ etale cyclic coverings of hyperbolic orbicurves and open immersions [with ∼ ∼ 2 / Z Z , Z /l Z , Gal( X /C ) ) normal image] of profinite groups; we have Gal( /C C = = → − ∼ ∼ Z / Z . l 2 /C and Gal( X ) × Gal( C /C → /C ) Gal( X ) = → − → − We shall refer to a hyperbolic orbicurve over that arises, up to Definition 1.1. k X ) constructed above isomorphism, as the hyperbolic orbicurve (respectively, C − → − → for some choice of l ,  ) (respectively, (1 as being (1 ,l -tors ) -tors ,l of type ). ± → − − − → −

39 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 39 → Remark 1.1.1. ” in the notation “ The arrow “ ”, “ C X )”, ”, “(1 ,l -tors − → → − → − − “archimedean, ordered labels ” may be thought of as denoting the “(1 ,l -tors ) ± → − − } 1 —onthe {± -orbits of ele- !  — i.e., determined by the ” ,... 2 1 , choice of Q ments of the quotient “ ” that appears in the definition of a “hyperbolic orbicurve ” given in [EtTh], Definition 2.1. of type (1 ,l -tors) ± X Remark 1.1.2. We observe that , C - k are completely determined, up to → − → − , ). isomorphism, by the data ( X/k, C is Corollary 1.2. (Characteristic Nature of Coverings) Suppose that k functorial group-theoretic algorithm an NF or an MLF. Then there exists a [cf. [AbsTopIII], Remark 1.9.8, for more on the meaning of this terminology] to reconstruct , Π ) Π (respectively, , Π Π C C X C → − together with the conjugacy classes of the decomposition group(s) determined by the ′ ′′ {  set(s) of cusps } ; {  } (respectively, {  } , Π )from ). Here, Π (respectively, C X − → − → the asserted functoriality is with respect to isomorphisms of topological groups; we Aut(Π Π Π (respectively, ) , Π , (respectively, )asasubgroupof reconstruct Π C X C X C → → − − ) ). Aut(Π C − → Proof. For simplicity, we consider the non-resp’d case; the resp’d case is entirely similar [but slightly easier]. The argument is similar to the arguments applied in ,Π ,and [EtTh], Proposition 1.8; [EtTh], Proposition 2.4. First, we recall that Π X X − → are slim [cf., e.g., [AbsTopI], Proposition 2.3, (ii)], hence embed naturally into Π C via the algorithms of ), and that one may recover the subgroup Δ Π ⊆ Aut(Π X X X → − → → − − [AbsTopI], Theorem 2.6, (v), (vi). Next, we recall that the algorithms of [AbsTopII], Corollary 3.3, (i), (ii) — which are applicable in light of [AbsTopI], Example 4.8 ], → Π [together with the natural inclusion Π ↪ — allow one to reconstruct Π X C C − → ⊆ Δ ⊆ Π .Inparticular, l may be recovered via as well as the subgroups Δ C C X 2 ]=[Δ 2. Next, let :Δ / ] · =[Δ ] :Δ :Δ ]=[Δ :Δ [Δ the formula l X X X C X X X X − → − → − → def ab ( Δ ⊆ may be recovered via the ⊗  Z /l Z )). Then Π Π =Ker(Δ H us set X X C X =Π · H . The conjugacy classes of the [easily verified] equality of subgroups Π X X → − ′′ 0 ′  decomposition groups of   in Π , , mayberecoveredasthe decomposition X groups of cusps [cf. [AbsTopI], Lemma 4.5, as well as Remark 1.2.2, (ii), below] , . Next, to reconstruct Π )=Π / nontrivial ⊆ Π is Π whose image in Gal( /X X X X C C − → → − /C )=Π / Π  of the surjection Gal( X it suffices to reconstruct the splitting X C prime to is / Π l = Gal( X/C ) determined by Gal( X /C )=Π ; but [since / Π Π X C X C 3!] this splitting may be characterized [group-theoretically!] as the unique splitting determined that stabilizes the collection of conjugacy classes of subgroups of Π X 0 ′ ′′ by the decomposition groups of  ,   , may be reconstructed Π ⊆ .NowΠ C C → − ∼ ∼ 2 / ) Z Z )( l ) Gal( by applying the observation that ( Z / 2 Z X ⊆ /C /C X Gal( ) = = → → − → − − is the unique maximal subgroup of odd index . Finally, the conjugacy classes of ′′ ′ ,  in Π mayberecoveredasthe decomposition  the decomposition groups of X groups of cusps [cf. [AbsTopI], Lemma 4.5, as well as Remark 1.2.2, (ii), below] not fixed [up to , but which are nontrivial is )=Π Π / /X whose image in Gal( X X X → − − →

40 40 SHINICHI MOCHIZUKI X /C conjugacy] by the outer action of Gal( / Π )=Π on Π . This completes the X C X © proof of Corollary 1.2. Remark 1.2.1. It follows immediately from Corollary 1.2 that ∼ ∼ Aut Z )=Gal( X /l Z /C )( 2 ( Z / X l Z ); Aut ) ( C )( /C )=Gal( C = = k k − → → − → → − − [cf. [EtTh], Remark 2.6.1]. Remark 1.2.2. The group-theoretic algorithm for reconstructing the decomposi- tion groups of cusps given [AbsTopI], Lemma 4.5 — which is based on the argument some minor, inessential given in the proof of [AbsAnab], Lemma 1.3.9 — contains . In light of the inaccuracies of this group-theoretic algorithm for the importance theory of the present series of papers, we thus pause to discuss how these inaccu- racies may be amended. ]ofthe second paragraph (i) The final portion [beginning with the third sentence of the proof of [AbsAnab], Lemma 1.3.9, should be replaced by the following text: may be recovered group-theoretically, given any finite ́ etale cov- r Since i erings V → X → Z i i i Z , is cyclic [hence Galois such that power of l ,over V ], of degree a i i → V is totally Z one may determine group-theoretically whether or not i i ], since this condition is easily verified Z ramified [i.e., at some point of i → V admit a to be equivalent to the condition that the covering Z i i , l etale of degree → W is finite ́ → V V ,where W → Z factorization i i i i i · r . Moreover, this group-theoreticity of the condition that a

41 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 41 in in Remark 1.2.2 are closely related to certain minor, inessential inaccuracies the theory of [CombGC]. Thus, it is of interest, in the context of the discussion of Remark 1.2.2, to pause to discuss how these inaccuracies may be amended. These inaccuracies arise in the arguments applied in [CombGC], Definition 1.4, (v), (vi), and [CombGC], Remarks 1.4.2, 1.4.3, and 1.4.4, to prove [CombGC], Theorem 1.6. These arguments are formulated in a somewhat confusing way and should be modified as follows: (i) First of all, we remark that in [CombGC], as well as in the following dis- “Galois” finite ́ connected . cussion, a etale covering is to be understood as being (ii) In the and second sentence of [CombGC], Definition 1.4, (v), the cuspidal purely totally ramified covering are in fact unnecessary cases of the notion of a nodal and may be deleted . Also, the terminology introduced in [CombGC], Definition descend is 1.4, (vi), concerning finite ́ and may be etale coverings that unnecessary . deleted (iii) The text of [CombGC], Remark 1.4.2, should be replaced by the following text: ′ →G be a Galois finite ́ etale covering of degree a positive power of Let G G is of pro-Σ PSC-type, Σ = { l } . Then one verifies immediately ,where l ′ ′ →G is cyclic ,then G →G G that, if we assume further that the covering cuspidally totally ramified is if and only if the inequality ′′ G

42 42 SHINICHI MOCHIZUKI def ′ ′′ ′ Π = · Π Π ⊆ A , then the cyclic finite ́ etale covering Π ⊆ G G G G ′′ ′ cuspidally totally ramified . →G is G of this necessity is immediate from the def- [Indeed, the characterization characterization follows by observing that sufficiency initions; the of this G is always finite ,the since the set of cusps of a finite ́ etale covering of implies that there exists a compatible system of cusps condition above of ′ that arise, each of which is by the action of A .] On stabilized the various G unramified verticial subgroups the other hand, in order to characterize the unr , it suffices — by considering stabilizers of vertices of underlying of Π G unr -coverings of G —togivea functorial char- semi-graphs of finite ́ etale Π G of the set of vertices of acterization [i.e., which may also be applied G unr -coverings of G ]. This may be done, for sturdy G ,as to finite ́ etale Π G unr unr for the abelianization of Π . For each vertex v of follows. Write M G G unr unr ⊆ [ v ] M for the image of of G G ,write the underlying semi-graph M G G unr unr associ- -conjugacy class of unramified verticial subgroups of Π the Π G G v ated to . Then one verifies immediately, by constructing suitable abelian unr -coverings G via suitable gluing operations [i.e., as in the proof of of Π G unr unr v ] ⊆ M [ determine a split M Proposition 1.2], that the inclusions G G injection ⊕ unr unr M M [ v ] ↪ → G G v unr-vert v ], whose image we denote by M [where ranges over the vertices of ⊆ G G unr . Now we consider elementary abelian quotients M G unr φ : M Q  G Q — i.e., where .We identify such quotients elementary abelian group is an kernels coincide order such quotients by means of the whenever their and relation of “domination” [i.e., inclusion of kernels]. Then one verifies im- unr  Q corresponds to a verticially : mediately that such a quotient φ M G purely totally ramified G if and only if there exists a vertex v covering of ′ unr unr ′ v of ]) = 0 for all vertices v v ]) = Q = φ ( M  [ [ v , M ( φ such that G of G G G . In particular, one concludes immediately that unr  Q whose restric- the elementary abelian quotients M φ : G unr-vert M tion to Q surjects onto and has the same kernel as the G quotient unr-vert unr unr M  [ M ]  M v [ v ] ⊗ F l G G G — where the first “  ” is the natural projection; the second “  ” is given by reduction modulo l — may be characterized as the [i.e., relative to the relation of domination] maximal quotients unr that corre- M among those elementary abelian quotients of G verticially purely totally ramified coverings of G . spond to Thus, since G is sturdy ,the set of vertices of G may be characterized as unr-vert unr . F  M [ v ] ⊗ the set of [nontrivial!] quotients M l G G

43 ̈ INTER-UNIVERSAL TEICHM 43 ULLER THEORY I (v) The text of [CombGC], Remark 1.4.4, should be replaced by the following text: { is } ,andthat G is of pro-Σ PSC-type, where Σ = l Suppose that G portion of Remark 1.4.3, cuspidal . Then, in the spirit of the noncuspidal we observe the following: One verifies immediately that the nodal edge-like [cf. Proposition 1.2, characterized as the maximal may be of Π subgroups G which satisfy the following isomorphic to Z A ⊆ (i)] closed subgroups Π l G : condition ′ for every characteristic open subgroup Π ⊆ Π ,ifwewrite G G ′ ′′ →G for the finite ́ etale coverings corresponding to →G G def ′ ′′ ′ Π finite ́ etale covering Π = A · Π cyclic ⊆ ⊆ Π , then the G G G G ′ ′′ is nodally totally ramified . →G G etale Here, we note further that [one verifies immediately that] the finite ́ ′ ′′ module-wise if and only if it is →G nodally totally ramified is G covering . nodal (vi) The text of the second paragraph of the proof of [CombGC], Theorem 1.6, should be replaced by the following text [which may be thought as being appended to the end of the first paragraph of the proof of [CombGC], Theorem 1.6]: Then the fact that α is group-theoretically cuspidal follows formally from the characterization of cuspidal edge-like subgroups given in Remark 1.4.3 cuspidally totally ramified cyclic finite ́ etale and the characterization of coverings given in Remark 1.4.2. (vii) The text of the final paragraph of the proof of [CombGC], Theorem 1.6, should be replaced by the following text [which may be thought of as a sort of “easy version” of the argument given in the proof of the implication “(iii) = ⇒ (i)” of [CbTpII], Proposition 1.5]: Sufficiency is immediate. On the Finally, we consider assertion (iii). other hand, necessity follows formally from the characterization of unram- ified verticial subgroups given in Remark 1.4.3 and the characterization of verticially purely totally ramified finite ́ etale coverings given in Remark 1.4.2.

44 44 SHINICHI MOCHIZUKI Section 2: Complements on Tempered Coverings 2, we discuss certain routine complements — which will be of § In the present tempered coverings of graphs use in the present series of papers — to the theory of , as developed in [SemiAnbd], 3 [cf. also the closely related theory § of anabelioids of [CombGC]]. ̂ ̂ ⊆ Let Σ, Σ; nonempty sets of prime numbers Σbe such that Σ G Σ PSC-type [cf. [CombGC], Definition 1.1, (i)], a semi-graph of anabelioids of pro- tp G whose underlying semi-graph we denote by .WriteΠ tempered funda- for the G mental group G [cf. the discussion preceding [SemiAnbd], Proposition 3.6, as of ̂ ̂ for the Σ [i.e., pro- Π well as Remark 2.5.3, (i), (T6), of the present paper] and G ̂ Σ quotient of the profinite] of G [cf. the discussion maximal pro- fundamental group preceding [SemiAnbd], Definition 2.2] — both taken with respect to appropriate choices of basepoints. Thus, since discrete free groups of finite rank inject into completions for any prime number l [cf., e.g., [RZ], Proposition 3.3.15], their pro- l natural injection [cf. [SemiAnbd], Proposition 3.6, (iii), as it follows that we have a ̂ Σ= ; the proof well as Remark 2.5.3, (ii), (E7), of the present paper, when Primes ̂ in the case of arbitrary Σ is entirely similar] tp ̂ ↪ → Π Π G G tp ̂ ̂ ̂ that we shall use to regard Π of subgroup Π Σ and completion Π as a as the pro- G G G tp . of Π G Next, let H be the semi-graph of anabelioids associated to a sub-semi-graph H ⊆ connected . One verifies immediately that the restriction of H to the G maximal subgraph [cf. the discussion at the beginning of [SemiAnbd], 1] of H coincides with the § restriction to the maximal subgraph of the underlying semi-graph of some semi- graph of anabelioids of pro-Σ PSC-type. That is to say, roughly speaking, up to the H “is” a semi-graph of anabelioids possible omission of some of the cuspidal edges, of pro-Σ PSC-type. In particular, since the omission of cuspidal edges clearly does ̂ not affect either the tempered or pro- Σ fundamental groups, we shall apply the notation introduced above for “ G ”to H .Wethusobtaina natural commutative diagram tp ̂ −→ Π Π H H ⏐ ⏐ ⏐ ⏐   tp ̂ Π −→ Π G G ̂ [cf. [SemiAnbd], Proposition 2.5, (i), when of [outer] inclusions Σ= Primes ;in light of the well-known structure of fundamental groups of hyperbolic Riemann ̂ surfaces of finite type, a similar proof may be given in the case of arbitrary Σ, i.e.,

45 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 45 by considering successive composites of finite ́ etale Galois coverings that restrict to over the closed edges and finite ́ trivial coverings etale abelian [Galois] coverings ob- abelian coverings ,whichwe ] of topological groups tained by together suitable gluing ̂ shall use to regard all of the groups in the diagram as subgroups of Π .Inpartic- G tp tp ̂ (respectively, )asthe decomposition subgroup in Π Π ular, one may think of Π H G H tp ̂ ̂ Π (respectively, - (respectively, ) [which is well-defined up to Π Π -)conjugacy] G G G associated to the sub-semi-graph H . central technical result The following result is the underlying the theory of the 2. § present Proposition 2.1. (Profinite Conjugates of Nontrivial Compact Sub- tp be a nontrivial In the notation of the above discussion, let ⊆ Λ Π groups) G tp 1 − ̂ Π ∈ γ , compact subgroup · γ ⊆ γ an element such that Π · Λ [or, equiva- G G tp tp 1 − γ lently, ⊆ Λ . · γ ∈ Π γ Π ]. Then · G G ̂ ̂ ̂ Σ semi-graph” associated to the universal pro- Write Σ ́ etale Γ for the “pro- Proof. tp ̂ ]; Γ for }⊆ G Π of { [i.e., the covering corresponding to the subgroup 1 covering G the “pro-semi-graph” associated to the universal tempered covering of G [i.e., the tp 1 }⊆ Π covering corresponding to the subgroup { ].Thus,wehaveanaturaldense G tp ̂ ̂ map Γ → Γ. Let us refer to a [“pro-”]vertex of Γ that occurs as the image of tp 1 − are . Since Λ, Λ · γ tempered as · compact subgroups of γ a[“pro-”]vertexofΓ tp Π , it follows from [SemiAnbd], Theorem 3.7, (iii) [cf. also [SemiAnbd], Example G 3.10, as well as Remark 2.5.3, (ii), (E7), of the present paper], that there exist tp ′ ′′ ′ ′′ ′ − 1 Λ verticial subgroups such that Λ ⊆ Λ Λ , γ · Λ · γ ⊆ Π ⊆ Λ , .Thus,Λ , G ′′ 1 − ′ ′ 1 ′′ − ̂ · { 1 }  = γ correspond to Λ · γ Γ; v ⊆ γ · Λ , · γ v tempered vertices ,so of Λ ⋂ 1 ′ − 1 ′′ ′′ ′ − · Λ γ  = { 1 } .SinceΛ are both verticial subgroups of , γ · Λ · ) γ · γ Λ ( ̂ , it thus follows either from [AbsTopII], Proposition 1.3, (iv), or from [NodNon], Π G ′′ ′ γ ̂ Γ are either v ,( of equal ) v Proposition 3.9, (i), that the corresponding vertices γ ′ ′′ is v is tempered , we thus conclude that ( ) v . In particular, since or adjacent tp ′ ′ γ .Thus, v tempered are tempered, so ) ∈ Π v , as desired. © ,( γ G C ”, “ Next, relative to the notation “ ” and related terminology concerning N commensurators normalizers discussed, for instance, in [SemiAnbd], § 0; [Com- and § bGC], 0, we have the following result. Proposition 2.2. (Commensurators of Decomposition Subgroups As- ̂ (re- sociated to Sub-semi-graphs) In the notation of the above discussion, Π H tp ̂ ̂ )is commensurably terminal in Π (respectively, [hence, also Π spectively, Π G G H tp tp ̂ in Π Π is commensurably terminal in Π . ]). In particular, G G G Proof. First, let us observe that by allowing, in Proposition 2.1, Λ to range over the nontrivial compact !] subgroup open subgroups of any verticial [hence, in particular, tp , we conclude from Proposition 2.1 that of Π G tp ̂ Π is commensurably terminal in Π G G

46 46 SHINICHI MOCHIZUKI H [cf. the discus- — cf. Remark 2.2.2 below. In particular, by applying this fact to tp sion preceding Proposition 2.1], we conclude that Π is commensurably terminal H ̂ Π in . Next, let us observe that it is immediate from the definitions that H tp tp tp ̂ tp ) (Π ⊆ ( ) ⊆ C C Π ) (Π ⊆ C Π H H H H Π ̂ ̂ Π Π G G G tp tp ̂ ̂ ̂ Π , ]. Σ completions of Π ,Π , respectively, as the pro- Π [where we think of G H H G ̂ On the other hand, by the evident pro- Σ analogue of [SemiAnbd], Corollary 2.7, (i) [cf. also the argument involving of abelian coverings in the discussion gluing ̂ ̂ commensurable .Thus,bythe ( Π Π )= preceding Proposition 2.1], we have C H H ̂ Π G tp ̂ of Π terminality in Π , we conclude that H H tp tp tp tp Π ) ⊆ (Π C (Π )=Π ⊆ C H H H H ̂ ̂ Π Π G H © — as desired. It follows immediately from the theory of [SemiAnbd] [cf., e.g., Remark 2.2.1. [SemiAnbd], Corollary 2.7, (i)] that, in fact, Propositions 2.1 and 2.2 can be proven for much more general semi-graphs of anabelioids than the sort of G that appears G in the above discussion. We leave the routine details of such generalizations to the interested reader. ̂ Recall that when Remark 2.2.2. , the fact that Σ= Primes tp ̂ Π is in Π normally terminal G G may also be derived from the fact that any nonabelian finitely generated free group [cf. [Andr ́ e], Lemma 3.2.1; [SemiAnbd], Lemma 6.1, (i)] in its normally terminal is profinite completion. In particular, the proof of the commensurable terminality of tp ̂ Π that is given in the proof of Proposition 2.2 may be thought of as a new in Π G G does not require one to invoke [Andr ́ e], Lemma of this normal terminality that proof conjugacy 3.2.1, which is essentially an immediate consequence of the rather difficult result given in [Stb1], Theorem 1. This relation of Proposition 2.1 to separability given in Theorem discrete analogue the theory of [Stb1] is interesting in light of the 2.6 below of [the “tempered version of Theorem 2.6” constituted by] Proposition 2.4 [which is essentially a formal consequence of Proposition 2.1]. def k/k = Gal( k an algebraic closure of a , G X ), k , MLF be an k Now let k of over k that admits stable reduction over the ring of integers O hyperbolic curve k k .Write tp tp Δ , Π X X tp ̂ Σ -tempered” quotients of the tempered fundamental groups π for the respective “ ( X ), 1 def tp π X X X , X ( § = ) [relative to suitable basepoints] of × e], k [cf. [Andr ́ 4; [Semi- k 1 k k tp tp is the quotient determined ( X )  Δ Anbd], Example 3.10]. That is to say, π 1 X k tp ( X )ontoex- by the intersection of the kernels of all continuous surjections of π 1 k tensions of a finite group of order a product [possibly with multiplicities] of primes

47 ̈ INTER-UNIVERSAL TEICHM 47 ULLER THEORY I tp tp tp ̂ )  Π π ∈ is the quotient of π Σ by a discrete free group of finite rank; ( ( X ) X 1 1 X tp tp ̂ π determined by the kernel of the quotient of ( Δ X  .Write ) Δ for the X 1 X k ̂ ̂ ̂ Σ quotient of the profinite] fundamental group X Σ [i.e., maximal pro- pro- of ; Π X k X by the subgroup of the for the quotient of the profinite fundamental group of ̂ X profinite fundamental group of Δ that determines the quotient . Thus, since X k completions for any prime l discrete free groups of finite rank inject into their pro- [cf., e.g., [RZ], Proposition 3.3.15], we have natural inclusions number l tp tp ̂ ̂ Π → ↪ Π , Δ ↪ → Δ X X X X [cf., e.g., [SemiAnbd], Proposition 3.6, (iii), as well as Remark 2.5.3, (ii), (E7), ̂ ̂ ̂ Σ pro- may be identified with the ]; of the present paper, when Primes Σ= Δ X tp tp ̂ ̂ Π . is generated by the images of Π ; Δ and of Δ completion X X X X p residue characteristic is not contained in Σ; of Now suppose that the k pro- of the above discussion is the semi-graph G that the semi-graph of anabelioids Σ of anabelioids associated to the geometric special fiber of the stable model X of X G [cf., e.g., [SemiAnbd], Example 3.10]; and that the sub-semi-graph H ⊆ over O k on G .Thus,wehave natural surjections by the natural action of G is stabilized k tp tp ̂ ̂ ;  Π Π  Δ Δ X G X G of topological groups. Corollary 2.3. (Subgroups of Tempered Fundamental Groups Associ- ated to Sub-semi-graphs) In the notation of the above discussion: (i) The closed subgroups def def tp tp tp tp ̂ ̂ ̂ ̂ tp Π ⊆ Δ × =Δ ; Δ Δ × ⊆ Δ = Π Δ X X X, H H X X H H X, Π ̂ Π G G commensurably terminal on are G . In particular, the natural outer actions of k tp tp ̂ ̂ Δ . natural outer actions of , determine on Δ Δ , G Δ k X X, H X H X, tp tp ̂ ̂ ̂ . ⊆ is equal to ⊆ Δ in Δ Δ Δ Δ of closure (ii) The X X X, H X X, H ̂ Σ (iii) Suppose that [at least] one of the following conditions holds: (a) contains ⋃ ̂ ̂ .Inparticular, is slim } ;(b) Σ Σ= Primes .Then ∈ Δ a prime number l/ { p H X, tp ̂ the natural outer actions of G Δ natural exact [cf. (i)] determine Δ , on X, k H H X, ̂ ; [cf. (ii); the slimness of Δ sequences of center-free topological groups X, H [AbsAnab], Theorem 1.1.1, (ii)] tp tp Δ → 1 → 1 Π → G → k H X, X, H ̂ ̂ 1 → Δ → → G Π 1 → H X, H X, k tp ̂ Π —where , so as to render the sequences exact. Π defined are H X, H X, (iv) Suppose that the hypothesis of (iii) holds. Then the images of the natural tp tp ̂ ̂ Π ↪ → Π are Π , . commensurably terminal → ↪ inclusions Π H X, X X X, H

48 48 SHINICHI MOCHIZUKI ⋂ tp tp ̂ ̂ =Δ Δ . Δ ⊆ (v) We have: Δ X X, H X H X, (vi) Let tp ̂ ⊆ (respectively, I ) Δ ⊆ Δ I X x x X x of X be an ξ for the cusp of the stable associated to a cusp inertia group . Write x . Then the following conditions are equivalent: model X corresponding to tp tp ̂ (respectively, Δ - (respectively, lies in a Δ -) conjugate of Δ (a) I X x X H X, ̂ ); Δ H X, X that is (b) ξ meets an irreducible component of the special fiber of con- H . in tained Proof. Assertion (i) follows immediately from Proposition 2.2. Assertion (ii) fol- lows immediately from the definitions of the various tempered fundamental groups observation G  F is a surjec- involved, together with the following elementary :If ̂ ̂  F tion of finitely generated free discrete groups, which induces a surjection G ̂ between the respective pro- Σ completions [so, since discrete free groups of finite l completions for any prime number l [cf., e.g., [RZ], rank inject into their pro- ̂ ̂ G Proposition 3.3.15], we think of G and as subgroups of F , respectively], and F def def ̂ ̂ ̂ ̂ Σ topol- in dense H G =Ker(  G  F F ), relative to the pro- )is =Ker( H then ̂ G . Indeed, let ι : F↪ → G be a section of the given surjection G  F ogy of [which } that converges, G is a sequence of elements of F exists since is free ]. Then if { g N ∈ i i ̂ ̂ ̂ ,toagivenelement G ∈ Σ topology of H , and maps to a sequence of h in the pro- ̂ ̂ ,to of F F [which necessarily converges, in the pro- Σ topology of } f elements { i i N ∈ 1 − ̂ } is ) f ( ι · g ], then one verifies immediately that F ∈ 1 identity element the { i ∈ N i i ̂ ̂ H G ,to h .This that converges, in the pro- a sequence of elements of Σ topology of completes the proof of the and hence of assertion (ii). observation Next, we consider assertion (iii). In the following, we give, in effect, two distinct ̂ , but requires one to assume that elementary :oneis Δ proofs of the slimness of H X, condition (a) holds; the other depends on the highly nontrivial theory of [Tama2] holds. If condition (a) holds, then and requires one to assume that condition (b) ⋃ def ∗ { l } . If condition (b) holds, but condition (a) does not hold [so =Σ let us set Σ ⋃ def ∗ ∗ ̂ ,and  ∈ Σ = Σ. Thus, in either case, p p } Primes =Σ Σ= { ], then let us set Σ ∗ ̂ ⊆ Σ. Σ Σ ⊆ ⋂ def ∗ ̂ ̂ for J = J normal open subgroup .Write be a ; J  J Δ Δ J Let ⊆ H X H X, ∗ ∗ ∗ ∗ quotient ; J J in ⊆ J J for the image of . Now suppose that Σ the maximal pro- H H ̂ be a vertex of the dual graph of the geometric commutes with J .Let v α Δ ∈ H H X, of of the covering X .Write J X determined by special fiber of a stable model X J J k ] associated ⊆ J for the decomposition group [well-defined up to conjugation in J J v ∗ ∗ ∗ . Then let us observe that in ⊆ J J for the image of J ; to J v v v ̂ J ) there exists an open subgroup ( † , Δ which is independent of J , v ⊆ X 0 , then for arbitrary v [and α ]asabove,itholds α such that if and J ⊆ J 0 ⋂ ∗ ∗ ∗ J infinite ( ⊆ J . )is nonabelian and J that v H Indeed, suppose that condition (a) holds. Now it follows immediately from the ̂ ̂ is ⊆ J ⊆ pro- Δ Π  Σ; in definitions that the image of the homomorphism J X G v

49 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 49 ⋂ ̂ ̂ J ⊆ Σ, and Ker( Δ  l Π J ) ⊆ J ⊆  , it follows that ∈ particular, since J H v v G X ⋂ ⋂ ∗ ∗ J J ,which , hence also J J of , surjects onto the maximal pro- l quotient J v v H v H completion of the fundamental group of a hyperbolic l is isomorphic to the pro- infinite Riemann surface, hence [as is well-known] is nonabelian [so we may and def ̂ = ]. Now suppose that condition (b) holds, but condition (a) does Δ J take X 0 not hold. Then it follows immediately from [Tama2], Theorem 0.2, (v), that, for J ⊆ J corresponds to an irreducible ,thenevery v ,if J an appropriate choice of 0 0 component that either maps to a point in or contains a node that maps to a X . In particular, it follows that for every choice of v , there exists smooth point of X that lies in Ker( J ⊆ J at least one pro- Σ , torsion-free, pro-cyclic subgroup F ⊆ v v ⋂ ∗ ̂ ̂ J .Thus,we ) ⊆ J J  and, moreover, maps injectively into Π ⊆ J Δ H G X v ⋂ ∗ ∗ J ; a similar statement holds when F is replaced by F↪ → J obtain an injection v H -conjugate of F . Moreover, it follows from the well-known structure of the any J v pro-Σ completion of the fundamental group of a hyperbolic Riemann surface [such ∗ ] that the image of the J topologically generate -conjugates of such a group F J as v v ⋂ ∗ ∗ J infinite . This completes the which is nonabelian and J a closed subgroup of v H proof of ( † ). Next, let us observe that it follows by applying either [AbsTopII], Proposition ∗ ̂ Δ 1.3, (iv), or [NodNon], Proposition 3.9, (i), to the various -conjugates in J of X ⋂ ⋂ ∗ ∗ ∗ ∗ J J † ) that the fact that α commutes with J as in ( α implies that fixes J v v H H . If condition (a) holds, then the fact that conjugation by α maximal pro- l v on the ⋂ ∗ ∗ J J [which, as we saw above, is a quotient of ]is trivial implies of J quotient v v H ” in the latter portion ⊆ D I [cf. the argument concerning the inertia group “ v v not only fixes , but also acts α v of the proof of [SemiAnbd], Corollary 3.11] that determined by v ; X trivially on the irreducible component of the special fiber of J v since † )is arbitrary ,wethusconcludethat α acts on the abelianization as in ( ab ∗ ∗ of J ) trivially unipotent automorphism of finite order , hence that α acts as a ( J ab ∗ )is ) ;since J as in ( † arbitrary , we thus conclude [cf., e.g., the proof of on ( J [Config], Proposition 1.4] that α is the identity element , as desired. Now suppose that condition (b) holds, but condition (a) does not hold. Then since J and v as in ( † arbitrary , we thus conclude again from [Tama2], Theorem 0.2, (v), that )are fixes not only v every closed point on the irreducible component of the α , but also v , hence that α acts trivially on this irreducible determined by X special fiber of J . Again since J and component as in ( † )are arbitrary , we thus conclude that v α identity element , as desired. This completes the proof of assertion (iii). is the In light of the exact sequences of assertion (iii), assertion (iv) follows immediately from assertion (i). Assertion (vi) follows immediately from a similar argument to the argument applied in the proof of [CombGC], Proposition 1.5, (i), by passing to pro-Σ completions. Finally, it follows immediately from the definitions of the various tempered fundamental groups involved that to verify assertion (v), it suffices to verify the following analogue of assertion (v) for a nonabelian finitely generated free discrete group G F ⊆ G , if we use the notation “ ∧ ” : for any finitely generated subgroup ⋂ ̂ ̂ to denote the pro- Σ completion, then F G = F . But to verify this assertion concerning G , it follows immediately from [SemiAnbd], Corollary 1.6, (ii), that we may assume without loss of generality that the inclusion ⊆ G admits a splitting F G  F ], in which F↪ → G  F is the identity on F [i.e., such that the composite

50 50 SHINICHI MOCHIZUKI ⋂ ̂ F ” follows immediately. This completes the case the desired equality “ G F = © proof of assertion (v), and hence of Corollary 2.3. arithmetic analogue Next, we observe the following of Proposition 2.1. Proposition 2.4. (Profinite Conjugates of Nontrivial Arithmetic Com- pact Subgroups) In the notation of the above discussion: tp ̂ be a nontrivial pro- Σ compact subgroup , γ ∈ Π an Δ Λ (i) Let ⊆ X X tp tp 1 − − 1 ⊆ γ Δ Δ [or, equivalently, Λ ⊆ · Λ element such that · γ γ · γ · ]. Then X X tp . ∈ γ Π X tp ̂ be a [nontrivial] compact ⊆ Π Λ Σ= Primes .Let (ii) Suppose that X 1 − ̂ G subgroup whose image in Λ ∈ γ , γ · Π · γ open is ⊆ an element such that X k tp tp tp 1 − Π [or, equivalently, γ ⊆ · · γ ]. Then γ ∈ Π Λ Π . X X X tp tp ̂ (respectively, Π )is commensurably terminal in Δ (respec- (iii) Δ X X X ̂ tively, Π ). X First, we consider assertion (i). We begin by observing Proof. that since [as is ̂ is , it follows strongly torsion-free well-known — cf., e.g., [Config], Remark 1.2.2] Δ X tp Δ that there exists a finite index characteristic open subgroup J ⊆ such that, if X for the pro- Σ semi-graph of anabelioids associated to the special fiber of we write G J ] of the finite ́ of k etale covering of O the stable model [i.e., over the ring of integers k ⋂ X × k determined by J ,then J completion Λhas nontrivial image in the pro- Σ k tp , hence in Π of of the abelianization J [since, as is well-known, our assumption G J tp induces an isomorphism between J  Π ∈ p/ that Σ implies that the surjection G J tp the pro-Σ completions of the respective abelianizations]. Since the quotient Π X tp surjects onto J is open of finite index in Δ G , we may assume without loss ,and k X ⋂ ̂ ̂ in of generality that lies in the closure J Π γ of J J Λhas .Since nontrivial X tp image in Π , it thus follows from Proposition 2.1 [applied to G ] that the image J G J tp ̂ ̂ ̂ via the natural surjection on pro- J  Π of Σ completions γ lies in Π . Since, G J G J tp ̂ vary ,Π by allowing J to (respectively, Π ) may be written as an inverse limit of X X tp tp ̂ ̂ ̂ the topological groups Π J  Π ) (respectively, Ker( Π J / Ker( / )), we  Π G X J X G J tp , as desired. thus conclude that [the original] γ lies in Π X Next, we consider assertion (ii). First, let us that it follows from a sim- observe ilar argument to the argument applied to prove Proposition 2.1 — where, instead of arithmetic analogue ,namely, applying [SemiAnbd], Theorem 3.7, (iii), we apply its [SemiAnbd], Theorem 5.4, (ii); [SemiAnbd], Example 5.6 [cf. also Remark 2.5.3, ̂ ̂ ̂ ∗ Ker(  / Π ) Δ (ii), (E5), (E7), of the present paper] — that the image of in γ Π X G X tp tp tp ̂ Ker(Δ ] Σ=  Π ), where [by invoking the hypothesis that Primes / lies in Π ∗ X X G ∗ we take G to be a semi-graph of anabelioids as in [SemiAnbd], Example 5.6, i.e., the semi-graph of anabelioids whose finite ́ etale coverings correspond to arbitrary admissible coverings of the geometric special fiber of the stable model X . Here, we note that when one applies either [AbsTopII], Proposition 1.3, (iv), or [NodNon],

51 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 51 tp G on Π to a Proposition 3.9, (i) — after, say, restricting the outer action of ∗ k G G that maps isomorphically onto of closed pro-Σ subgroup of the inertia group I k k ′ γ ′′ ”, “( v — to the vertices “ ) v ”, one may only I the maximal pro-Σ quotient of k conclude that these two vertices either ,are adjacent ,or admit a common coincide γ ′ ) ” v temperedness ; but this is still sufficient to conclude the adjacent vertex of “( ′′ ”. Now [just as in the proof of assertion (i)] by applying [the evident from that of “ v tp tp tp to the quotients Π analogue of] this observation  / Ker( J  Π Π )—where ∗ G X X J tp ∗ J Δ ⊆ semi-graph of G is the is a finite index characteristic open subgroup, and J X anabelioids whose finite ́ etale coverings correspond to arbitrary admissible coverings k × of the finite ́ etale covering of X of the special fiber of the stable model over O k k tp , as desired. — we conclude that γ J ∈ Π determined by X Finally, we consider assertion (iii). Just as in the proof of Proposition 2.2, the tp ̂ commensurable terminality of Δ in follows immediately from assertion (i), Δ X X by allowing, in assertion (i), Λ to range over the open subgroups of a pro-Σ Sylow tp associated to an irreducible component subgroup of a decomposition group ⊆ Δ X tp ̂ . The commensurable terminality of Π X of the special fiber of in then follows Π X X tp ̂ immediately from the commensurable terminality of Δ . Δ © in X X ̂ Primes , the proof given above of Proposition Remark 2.4.1. Σ= Thus, when e], Corollary 6.2.2 [cf. also [SemiAnbd], Lemma of [Andr ́ 2.4, (iii), yields a new proof of [Andr ́ e], Lemma 3.2.1, hence also of [Stb1], 6.1, (ii), (iii)] which is independent Theorem 1 [cf. the discussion of Remark 2.2.2]. Corollary 2.5. (Profinite Conjugates of Tempered Decomposition and Inertia Groups) In the notation of the above discussion, suppose further that ̂ ̂ (respectively, inertia in Σ= Π Primes . Then every decomposition group X ̂ (respectively, to a cusp of ) associated to a closed point or cusp of X Π group in X tp tp X )iscontainedin Π if and only if it is a decomposition group in Π (respectively, X X tp ) associated to a closed point or cusp of X (respectively, to a Π inertia group in X tp ̂ X cusp of Π ). Moreover, a Π contains a decomposition group in -conjugate of X X tp tp (respectively, inertia group in Π X ) associated to a closed point or cusp of Π X X tp . (respectively, to a cusp of X Π ) if and only if it is equal to X tp tp D Let Proof. Π ⊆ be the decomposition group in Π associated to a closed x X X ⋂ def tp ̂ Δ = D Π . Then the decomposition groups of I X of x point or cusp ; X x x X ̂ are precisely the associated to Π x -conjugates of D ; the decomposition groups x X tp tp of Π associated to x are precisely the Π .Since D -conjugates of D is compact x x X X , it thus follows from Proposition 2.4, and surjects onto an open subgroup of G k tp ̂ ifandonlyifitis,infact,a D is contained in Π -conjugate of Π (ii), that a x X X tp tp ̂ Π -conjugate of , and that a Π if and only if -conjugate of Π D contains D x x X X X tp ∼ ̂ Z ], is a cusp of X [so I . In a similar vein, when x it is, in fact, equal to Π = x X it follows — i.e., by applying Proposition 2.4, (i), to the unique maximal pro-Σ tp ̂ I subgroup of ifandonlyifitis, —thata Π I is contained in Π -conjugate of x X x X tp tp ̂ in fact, a Π -conjugate of Π , and that a if and Π contains I I -conjugate of X x x X X tp . This completes the proof of Corollary 2.5. © only if it is, in fact, equal to Π X

52 52 SHINICHI MOCHIZUKI The content of Corollary 2.5 may be regarded as a sort of [very Remark 2.5.1. of anabelian geometry — i.e., as the weak!] version of the “Section Conjecture” assertion that certain sections of the tempered fundamental group [namely, those that arise from geometric sections of the profinite fundamental group] are geometric as sections of the tempered fundamental group. This point of view is reminiscent of the point of view of [SemiAnbd], Remark 6.9.1. Perhaps one way of summarizing this circle of ideas is to state that one may think of (i) the classification of maximal compact subgroups of tempered fundamental groups given in [SemiAnbd], Theorem 3.7, (iv); [SemiAnbd], Theorem 5.4, (ii) [cf. also Remark 2.5.3, (ii), (E5), (E7), of the present paper], or, for that matter, “any finite group acting on a tree [without (ii) the more elementary fact that inversion] fixes at least one vertex” [cf. [SemiAnbd], Lemma 1.8, (ii)] from which these results of [SemiAnbd] are derived as a sort of combinatorial version of the Section Conjecture . Remark 2.5.2. Ultimately, when we apply Corollary 2.5 in [IUTchII], it will only be necessary to apply the portion of Corollary 2.5 that concerns inertia groups , i.e., the portion whose proof only requires the use of Proposition 2.4, of cusps (i), which is essentially an immediate consequence of Proposition 2.1. That is to say, the theory developed in [IUTchII] [and indeed throughout the present series of papers] will never require the application of Proposition 2.4, (ii), i.e., whose proof depends on a slightly more complicated version of the proof of Proposition 2.1. Remark 2.5.3. In light of the importance of the theory of [SemiAnbd] in the § 2, we pause to discuss certain minor oversights on the part of the author present in the exposition of [SemiAnbd]. (i) Certain pathologies occur in the theory of tempered fundamental groups if one does not impose suitable countability hypotheses. In order to discuss these countability hypotheses, it will be convenient to introduce some terminology as follows: (T1) We shall say that a tempered group is Galois-countable if its topol- ogy admits a countable basis. We shall say that a connected temperoid is if it arises from a Galois-countable tempered group. Galois-countable Galois-countable We shall say that a temperoid is if it arises from a col- lection of Galois-countable connected temperoids. We shall say that a connected quasi-temperoid is Galois-countable if it arises from a Galois- countable connected temperoid. We shall say that a quasi-temperoid is if it arises from a collection of Galois-countable connected Galois-countable quasi-temperoids. (T2) We shall say that a semi-graph of anabelioids G is Galois-countable if it is countable, and, moreover, admits a countable collection of finite ́ etale , H→G etale covering →G} such that for any finite ́ {G coverings ∈ i i I →G G ∈ I such that the base-changed covering H× there exists an i i G i splits over the constituent anabelioid associated to each component of [the . underlying semi-graph of] G i

53 ̈ INTER-UNIVERSAL TEICHM 53 ULLER THEORY I G is if it (T3) We shall say that a semi-graph of anabelioids strictly coherent is coherent [cf. [SemiAnbd], Definition 2.3, (iii)], and, moreover, each of c of [the underlying semi- the profinite groups associated to components graph of] [cf. the final sentence of [SemiAnbd], Definition 2.3, (iii)] is G topologically generated by N , for some positive integer N that generators of c . In particular, it follows that if G is finite and coherent , independent is strictly coherent then it is . , countable semi- strictly coherent (T4) One verifies immediately that every Galois-countable . graph of anabelioids is (T5) One verifies immediately that if, in [SemiAnbd], Remark 3.2.1, one as- X is sumes in addition that the temperoid , then it follows Galois-countable temp ( X )is well-defined that its associated tempered fundamental group π 1 and . Galois-countable (T6) One verifies immediately that if, in the discussion of the paragraph preceding [SemiAnbd], Proposition 3.6, one assumes in addition that the is Galois-countable , then it follows that its semi-graph of anabelioids G temp temp ( ( ) G )and temperoid B G π tempered fundamental group associated 1 are well-defined and Galois-countable . Here, we note that, in (T5) and (T6), the Galois-countability assumption is nec- essary in order to ensure that the index sets of “universal covering pro-objects” tempered fundamental group may to be taken to implicit in the definition of the . This countability of the index sets involved implies that the various be countable compatible sys- objects that constitute such a universal covering pro-object admit a obstruction to the existence of such a compatible tem of basepoints , i.e., that the 1 lim ” system — which may be thought of as an element of a sort of “nonabelian R ← − vanishes — . In order to define the tempered fundamental group in an intrinsi- cally meaningful fashion, it is necessary to know the existence of such a compatible system of basepoints. (ii) The of the omission of Galois-countability hypotheses in [SemiAnbd], effects § 3 [cf. the discussion of (i)], on the remainder of [SemiAnbd], as well as on subse- quent papers of the author, may be summarized as follows: (E1) First of all, we observe that all topological subquotients of absolute Galois groups of fields of countable cardinality are Galois-countable . (E2) Also, we observe that if k is a field whose absolute Galois group is Galois- countable ,and is a nonempty open subscheme of a connected proper U -scheme that arises as the underlying scheme of a log scheme that is k X k [where we regard Spec( log smooth over ) as equipped with the trivial k log structure], and whose interior is equal to U , then the tamely ramified arithmetic fundamental group of U [i.e., that arises by considering finite ́ U with tame ramification over the divisors that lie in etale coverings of the complement of U in X ] is itself Galois-countable [cf., e.g., [AbsTopI], Proposition 2.2]. (E3) Next, we observe, with regard to [SemiAnbd], Examples 2.10, 3.10, and 5.6, that the tempered groups and temperoids that appear in these Examples are Galois-countable [cf. (E1), (E2)], while the semi-graphs of

54 54 SHINICHI MOCHIZUKI strictly coherent [cf. item anabelioids that appear in these Examples are .Inparticular, Galois-countable (T3) of (i)], hence [cf. item (T4) of (i)] no effect on the theory of objects discussed in these Examples. there is (E4) It follows immediately from (E3) that there is no effect on [SemiAnbd], 6. § (E5) It follows immediately from items (T3), (T4) of (i), together with the assumptions of coherence in the discussion of the para- finiteness and graph immediately preceding [SemiAnbd], Definition 4.2, the assumption of coherence in [SemiAnbd], Definition 5.1, (i), and the assumption of [SemiAnbd], Definition 5.1, (i), (d), that there is no effect on [SemiAnbd], 4, § 5. [Here, we note that since the notion of a tempered covering § of a semi-graph of anabelioids is only defined in the case where the semi-graph countable , it is implicit in [SemiAnbd], Proposition 5.2, of anabelioids is and [SemiAnbd], Definition 5.3, that the semi-graphs of anabelioids under countable .] consideration are on [SemiAnbd], (E6) There is 1, § 2, or the Appendix of [SemiAnbd], no effect § since are never discussed in these portions tempered fundamental groups of [SemiAnbd]. (E7) In the Definitions/Propositions/Theorems/Corollaries of [SemiAnbd] that are numbered 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, one must assume that all tempered groups, temperoids, and semi-graphs of anabelioids that appear are . On the other hand, it follows immediately Galois-countable from (E1), (E2), and (E3) that there is no effect on the remaining portions of [SemiAnbd], § 3. (E8) In [QuCnf] and [FrdII], one must assume that all tempered groups and [quasi-]temperoids that appear are . Galois-countable no effect (E9) There is on any papers of the author other than [SemiAnbd] and the papers discussed in (E8). (iii) The assertion stated in the second display of [SemiAnbd], Remark 2.4.2, is false as stated. [The automorphisms of the semi-graphs of anabelioids in [Semi- “Dehn twists” constitute a well-known coun- Anbd], Example 2.10, that arise from assertion terexample to this assertion should be replaced by the following .] This assertion slightly modified version of this : completely determine the isomorphism The isomorphism classes of the φ v , up to composi- φ , as well as each isomorphism classofeachofthe φ e b tion with an automorphism of the composite 1-morphism of anabelioids that arises from an automorphism of the 1-morphism of →H →H G f w e →H . anabelioids G e f Also, in the following this assertion [as well as the various places where discussion this discussion is applied, i.e., [SemiAnbd], Remark 3.5.2; the second paragraph of [SemiAnbd], § 4; [SemiAnbd], Definition 5.1, (iv)], it is necessary to assume further that the semi-graphs of anabelioids that appear satisfy the condition that every edge abuts to at least one vertex .

55 ̈ INTER-UNIVERSAL TEICHM 55 ULLER THEORY I Galois ” at the end of the first sentence of the proof of (iv) The phrase “is ob- Galois [SemiAnbd], Proposition 3.2, should read “is a countable coproduct of jects”. (v) In the first sentence of [SemiAnbd], Definition 3.5, (ii), the phrase “Suppose splits that” should read “Suppose that each connected component of”; the phrase “ splits the restriction of this connected component the restriction of” should read “ of”. (vi) In order to carry out the argument stated in the proof of [SemiAnbd], Proposition 5.2, (i), it is necessary to the conditions (c) and (d) of strengthen [SemiAnbd], Definition 5.1, (i), as follows. This strengthening of the conditions (c) either on the remainder and (d) of [SemiAnbd], Definition 5.1, (i), has no effect of [SemiAnbd] or on subsequent papers of the author. Suppose that G is as in [SemiAnbd], Definition 5.1, (i). Then we begin by making the following observation : is finite .Then G admits a cofinal, countable collection (O1) Suppose that G i →G} ,eachofwhichis {G connected finite ́ of etale Galois coverings I i ∈ characteristic G ) [i.e., any pull-back of the covering via an element of Aut( is isomorphic to the original covering]. [For instance, one verifies immedi- finiteness and coherence of ately, by applying the , that such a collection G of coverings may be obtained by considering, for a positive integer, the n of all connected finite ́ composite n .] We etale Galois coverings of degree ≤ may assume, without loss of generality, that this collection of coverings ̃ projective system arises from a G . Thus, we obtain a , which we denote by natural exact sequence ̃ ̃ 1 Gal( / G ) −→ Aut( −→ G 1 G ) −→ Aut( G ) −→ G / ̃ G / G )” for the group of pairs of compatible auto- — where we write “Aut( ̃ G and G . morphisms of This observation (O1) has the following immediate consequence: (O2) Suppose that we are in the situation of (O1). Consider, for i ∈ I ,the finite index normal subgroup i ̃ ̃ / G ) ⊆ ( Aut( G / G ) G Aut ̃ G / G ) that induce the identity of elements of Aut( automorphism on the i i i ). Then one of G ,aswellasonGal( G / G G underlying semi-graph semi-graph of anabelioids ; verifies immediately [from the definition of a cf. also [SemiAnbd], Proposition 2.5, (i)] that the intersection of the i i ̃ ̃ G / G ), for i ∈ I ,is= { 1 } .Thus,theAut ( ( ,de- G / G ), for i ∈ I Aut ̃ on Aut( termine a G / G natural profinite topology ) and hence also on the quotient Aut( G ), which is easily seen to be compatible with the profinite ̃ ̃ G / G ) and, moreover, independent of the choice of topology on Gal( G . The of the condition (c) of [SemiAnbd], Definition 5.1, (i), that we new version wish to consider is the following: new H ) The action of H on G is trivial; the resulting homomorphism → (c Aut( G [ c ]), where c ranges over the components [i.e., vertices and edges]

56 56 SHINICHI MOCHIZUKI continuous G of [i.e., relative to the natural profinite group topology ,is ])]. defined in (O2) on Aut( c [ G new It is immediate that (c ) implies (c). Moreover, we observe in passing that: topologically finitely generated (O3) In fact, since H is [cf. [SemiAnbd], Defi- nition 5.1, (i), (a)], it holds [cf. [NS], Theorem 1.1] that every finite index new ) in fact . Thus, the conditions (c) and (c H H is open in subgroup of hold automatically . of the condition (d) of [SemiAnbd], Definition 5.1, (i), that we The new version wish to consider is the following: new ∗ (d )Thereisa of components [i.e., vertices and edges] of finite C set G ∗ ∗ ∈ C and an c of G , there exists a c such that for every component ∼ ∗ compatible c [ ]thatis →G ] c G of semi-graphs of anabelioids isomorphism [ on both sides. with the action of H new It is immediate that (d ) implies (d). The reason that, in the context of the proof of [SemiAnbd], Proposition 5.2, (i), it is necessary to consider the stronger new new )and(d ) is as follows. It suffices to show that, given a connected (c conditions ′ →G , after possibly replacing H by an open subgroup of finite ́ etale covering G ′ that satisfies the conditions of H lifts to an action on G , the action of H on G on to an action H [SemiAnbd], Definition 5.1, (i). Such a lifting of the action of G ′ vertices of G follows in a straightforward that lies over the G on the portion of original conditions (a), (b), (c), and (d). On the other hand, manner from the H in order to conclude that such a lifting is [after possibly replacing by an open subgroup of H ] compatible with the gluing conditions arising from the structure of ′ over the edges of G , it is necessary to assume further that the “component-wise G new new hold. This ), ( d “vertex-wise conditions (c), (d)” )” of the original c versions ( issue is closely related to the issue discussed in (iii) above. Finally, we observe that Proposition 2.4, Corollary 2.5 admit the following e], Lemma discrete analogues , which may be regarded as generalizations of [Andr ́ = = G is free]; [EtTh], Lemma H 3.2.1 [cf. Theorem 2.6 below in the case where F 2.17, (i). F be Theorem 2.6. (Profinite Conjugates of Discrete Subgroups) Let ⊆ G such that G is either a F a group that contains a subgroup of finite index orientable surface group [i.e., a or an free discrete group of finite rank ≥ fundamental group of a compact orientable topological surface of genus ]; H ⊆ F 2 an infinite subgroup. Since is residually finite [cf., e.g., [Config], Proposition 7.1, F ̂ ̂ profinite completion ⊆ H,G F ,where F (ii)], we shall write denotes the ⊆ of F ̂ F .Let γ ∈ F be an element such that 1 − 1 − ⊆ F F [or, equivalently, H ⊆ γ · · γ ]. H · γ γ · ⋂ def − 1 − 1 ,for G .Then γ ∈ F · N δ · ( H H ) , i.e., γ · H · · γ = H = δ H Write G G G G ̂ F . F ∈ γ ,then nonabelian is F ∈ δ some . If, moreover, H G Let us first consider the case where H Proof. is abelian . In this case, it follows G is cyclic . Thus, by applying Lemma 2.7, from Lemma 2.7, (iv), below, that H G

57 ̈ INTER-UNIVERSAL TEICHM 57 ULLER THEORY I G (ii), it follows that by replacing by an appropriate finite index subgroup of G , ab → G ↪ is a  G H we may assume that the natural composite homomorphism G ̂ ( H H )= , N . In particular, by Lemma 2.7, (v), we conclude that split injection G G ̂ G ̂ ̂ H G .Next, for the closure of G of in the profinite completion wherewewrite H G G F on the left by an appropriate element of γ let us observe that by multiplying ,we ⋂ − 1 ̂ ̂ F γ · G = G .Next,letus ⊆ · may assume that γ ∈ H G .Thus,wehave γ G G . Indeed, this is precisely the content of [Stb1], is conjugacy separable recall that ; [Stb2], Theorem 3.3, when G is an orientable surface Theorem 1, when G is free 1 − − 1  · H for ·  γ = · .Since G is conjugacy separable, it follows that γ · H group G G ̂ ∈ G ,so γ ∈ G · some  N ( ), as desired. This completes )= G · H H ( H F · N ⊆ G G G ̂ ̂ F G abelian . is H the proof of Theorem 2.6 when G H Thus, let us assume for the remainder of the proof of Theorem 2.6 that is G nonabelian . Then, by applying Lemma 2.7, (iii), it follows that, after replacing G by , we may assume that there exist elements an appropriate finite index subgroup of G ab G that generate a free abelian subgroup of rank two M ⊆ such that ∈ x, y H G ab H ⊆ .Write splits for the subgroups generated, H ,H G → M↪ the injection x G y ̂ ̂ ̂ ; respectively, by H x and y H ⊆ , G for the respective closures of H , H .Then x x y y ̂ ̂ N by Lemma 2.7, (v), we conclude that H N )= .Next,letus H H , ( )= H ( y x y x ̂ ̂ G G observe that by multiplying γ on the left by an appropriate element of F ,wemay ⋂ − 1 ̂ ̂ assume that · H ∈ γ G .Thus,wehave γ .Inparticular,by ⊆ · G = G γ F G applying the portion of Theorem 2.6 that has already been proven to the subgroups ̂ ̂ . ⊆ G , we conclude that γ ∈ G · N · G ( H )= )= G · H H ,H , γ ∈ G · N H ( H y x x y x y ̂ ̂ G G ab ̂ G Thus, by projecting to is of , we conclude M rank two , and applying the fact that that G , as desired. This completes the proof of Theorem 2.6. © γ ∈ ,then abelian is Note that in the situation of Theorem 2.6, if Remark 2.6.1. H G — unlike the tempered case discussed in Proposition 2.4! — it is not necessarily − 1 · . γ · F γ thecasethat F = (Well-known Properties of Free Groups and Orientable Lemma 2.7. ̂ G be a group as in Theorem 2.6. Write Let Surface Groups) for the profinite G completion of G .Then: (i) Any subgroup of G generated by two elements of G is free . (ii) Let x be an element  =1 . Then there exists a finite index subgroup ∈ G ab nontrivial image such that x ∈ G ⊆ ,and x has G in the abelianization G G 1 1 1 of G . 1 G x, y be noncommuting elements of G . Then there exists a (iii) Let ∈ n n ,and and a positive integer n such that x G ,y ⊆ ∈ G G finite index subgroup 1 1 n n ab generate a in the abelianization and y of G free abelian G the images of x 1 1 . subgroup of rank two G is cyclic . (iv) Any abelian subgroup of ̂ ̂ T (v) Let ⊆ G be a closed subgroup such that there exists a continuous surjec- ∼ ̂ ̂ ̂ ̂ ̂ Z .Then → T is Z that induces an isomorphism G T  tion of topological groups ̂ normally terminal in G .

58 58 SHINICHI MOCHIZUKI ̂ ̂ . Write (vi) Suppose that N ⊆ is G for the kernel of the natural nonabelian G ab ab ̂ ̂ ̂ ̂ ̂ G G . Then the centralizer Z to the abelianization of ( N ) surjection G G  ̂ G ̂ ̂ G N trivial . in of is be an automorphism of the profinite group (vii) In the notation of (vi), let α ̂ ̂ that preserves and restricts to the identity on the subgroup .Then G is the α N ̂ G identity automorphism of . is free G First, we consider assertion (i). If Proof. , then assertion (i) follows from the well-known fact that any subgroup of a free group is free. If G is an orientable surface group , then assertion (i) follows immediately — i.e., by consid- covering of a compact surface that corresponds to an infinite ering the noncompact G subgroup of of the sort discussed in assertion (i) — from a classical result index due to Johansson [cf. concerning the fundamental group of a noncompact surface [Stl], p. 142; the discussion preceding [FRS], Theorem A1]. This completes the G is residually finite proof of assertion (i). Next, we consider assertion (ii). Since [cf., e.g., [Config], Proposition 7.1, (ii)], it follows that there exists a finite index to be the G ⊆ G such that x  ∈ G . Thus, it suffices to take normal subgroup G 0 0 1 and x . This completes the proof of assertion (ii). generated by G G subgroup of 0 Next, we consider assertion (iii). By applying assertion (i) to the subgroup J generated by x and y , it follows from the fact that x and y of noncommuting G are a b is a G 2, hence that x that elements of free group of rank J  = 1, for all · y a, b ) ∈ Z × Z such that ( a, b )  =(0 , 0). Next, let us recall the well-known fact ( G that the abelianization of any finite index subgroup of .Thus, is torsion-free and , we conclude that there exists a finite index x y by applying assertion (ii) to m m m m such that x y ⊆ G ∈ G and a positive integer ,and x m and ,y G subgroup 0 0 ab mb ma . Now suppose that have of G G in the abelianization x nontrivial image · y 0 0 ab kernel of the natural surjection G Z  G lies in the × for some ( a, b ) ∈ Z 0 0 =(0 , 0). Since G is  , and [as we observed above] a, b such that ( ) residually finite mb ma · = 1, it follows, by applying assertion (ii) to G y , that there exists a finite  x 0 ⊆ G and a positive integer n that is divisible by m such that G index subgroup 0 1 n n ma mb ab ma mb ab ∈ G ,x , and the image of x ,y · y y in G · G is nontrivial .Since x 1 1 1 ab nb na nontrivial · x in G , it thus follows that the image of is is y .On torsion-free 1 ab ab → G ,wethus the other hand, by considering the natural homomorphism G 0 1 n n ab y in G and generate a free abelian subgroup of conclude that the images of x 1 rank two , as desired. This completes the proof of assertion (iii). Next, we consider assertion (iv). By assertion (i), it follows that any abelian generated by two elements is .Inparticular,we , hence cyclic subgroup of G free of G is equal to the union of the groups conclude that any abelian subgroup J .On G ⊆ G of cyclic subgroups of ⊆ ... ⊆ G G that appear in some chain 2 1 , it follows that the other hand, by applying assertion (ii) to some generator of G 1 n ⊆ and a positive integer n such that G G G there exists a finite index subgroup 0 0 j n ab =1 , 2 ,... , and, moreover, G for all . Thus, by has nontrivial image in G j 0 1 ab of the chain considering the image in [the finitely generated abelian group] G 0 n n , we conclude that this chain, hence also the ⊆ ⊆ G ... G of cyclic subgroups 2 1 , as desired. This cyclic is ⊆ G J ⊆ ... ,must terminate .Thus, G original chain 2 1 completes the proof of assertion (iv).

59 ̈ INTER-UNIVERSAL TEICHM 59 ULLER THEORY I ̂ ̂ G Next, we consider assertion (v). By considering the surjection Z  , we con- ̂ ̂ ̂ )of T in ( G is equal to the centralizer T clude immediately that the normalizer N ̂ G ̂ ̂ ̂ ̂ ̂ Z in Z G .If T )of , then it follows immediately that, for some prime ( ( T )  = T T ̂ ̂ G G ̂ ̂ l ⊆ Z ( T ) containing the pro- number l , there exists a closed [abelian] subgroup T 1 ̂ G ̂ ̂ portion of T such that there exists a continuous surjection Z ×  Z T whose ker- 1 l l l -cohomological × Z ). In particular, one computes easily that the · Z nel lies in ( l l l ̂ ̂ ̂ ≥ 2. On the other hand, since ,it T G is of infinite index in is T of dimension 1 1 ̂ ̂ ̂ ⊆ G such that of G follows immediately that there exists an open subgroup G 1 ̂ ̂ ̂ G whose ⊆ φ : , and, moreover, there exists a continuous surjection G Z  T 1 l 1 1 ̂ ̂ may be com- T . In particular, since the cohomology of T φ kernel Ker( ) contains 1 1 ̂ ̂ , G puted as the direct limit of the cohomologies of open subgroups of containing T 1 φ , together with the well-known struc- it follows immediately from the existence of ̂ -cohomological dimension ture of the cohomology of open subgroups of , that the l G ̂ is 1, a contradiction. This completes the proof of assertion (v). of T 1 ⊆ G for the kernel of the natu- N Next, we consider assertion (vi). Write ab ab G  ral surjection G G G . It follows immediately to the abelianization of of a free group or an orientable sur- “tautological universal property” from the face group [i.e., regarded as the quotient of a free group by a single relation] that not cyclic , hence by assertion (iv), that N is nonabelian N is . Thus, by asser- ⊆ G equipped with a surjection G tion (iii), there exist a finite index subgroup 1 ⋂ , )=(1  Z × Z and elements x, y ∈ N G 0) and x such that β ( : β G 1 1 y , 1). In particular, it follows from assertion (v) that the closed subgroups β ( )=(0 ̂ ̂ ̂ T , respectively, are normally termi- , ⊆ topologically generated by x and y G T y x ̂ ̂ nal in the profinite completion G G G of ⊆ . But this implies formally that 1 1 ⋂ ⋂ ⋂ ̂ ̂ ̂ ̂ ̂ ̂ Z T Z 1 { ) ( ( T ) } Z N G ( [where the last equal- T ) ⊆ ⊆ T = x 1 y x y ̂ ̂ ̂ G G G 1 1 ̂ ̂ ̂ × induced by Z Z β ]. Since [as  ity follows from the existence of the surjection G 1 ̂ are torsion-free ,we G is well-known] the abelianizations of all open subgroups of ̂ thus conclude that Z 1 N )= { ( } , as desired. This completes the proof of assertion ̂ G − 1 ̂ ̂ ̂ ∈ N ], then [so x · y · x x (vi). Finally, we consider assertion (vii). If G , y ∈ ∈ N − − 1 1 − 1 − 1 ( x · y · x = .Wethus )= α ( x ) · α ( y ) · α ( x ) ) α = α ( x ) · y · α ( x x x y · · − 1 ̂ ( ( α x )= ∈ Z , i.e., that . } x N )= { 1 ( conclude from assertion (vi) that x · ) x α ̂ G This completes the proof of assertion (vii). © Corollary 2.8. (Subgroups of Topological Fundamental Groups of Com- for . Write hyperbolic curve over C Z Π plex Hyperbolic Curves) be a Let Z ̂ for the profinite completion of ; Z the usual topological fundamental group of Π Z !]; H ⊆ Π be an infinite subgroup [such as a cuspidal inertia group .Let Π Z Z ̂ an element such that γ ∈ Π Z 1 − 1 − γ · · γ H [or, equivalently, H ⊆ γ ⊆ Π · Π ]. · γ Z Z 1 − − 1 ,forsome · N γ . If, moreover, δ ( H ) , i.e., Π · H · γ ∈ δ = δ · H · Π ∈ γ Then Z Z ̂ Π Z H is nonabelian ,then γ ∈ Π . Z Remark 2.8.1. Corollary 2.8 is an immediate consequence of Theorem 2.6. In fact, in the present series of papers, we shall only apply Corollary 2.8 in the case

60 60 SHINICHI MOCHIZUKI where Z is non-proper ,and H is a cuspidal inertia group . Inthiscase,theproof of Theorem 2.6 may be simplified somewhat, but we chose to include the general version given here, for the sake of completeness.

61 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 61 -Hodge Theaters Section 3: Chains of Θ Θ . Each “Θ-Hodge 3, we construct -Hodge theaters” chains of “ In the present § theater” is to be thought of as a sort of miniature model of the conventional that surrounds the theta function scheme-theoretic arithmetic geometry . [cf. [FrdI]; [FrdII]; Frobenioids This miniature model is formulated via the theory of § 4, § 5]. On the other hand, the link [cf. Corollary 3.7, (i)] between [EtTh], § 3, , i.e., it lies outside purely Frobenioid-theoretic adjacent members of such chains is the framework of ring theory/scheme theory. It is these chains of Θ-Hodge theaters starting point of the theory of the present series of papers. that form the Θ -data any collection of data Definition 3.1. We shall refer to as initial bad ,l,C , F/F, X V , V ) , ( F mod K that satisfies the following conditions: √ (a) is a number field such that F ∈ F ; F is an algebraic closure of F . − 1 def = Gal( F/F ). Write G F X (b) once-punctured elliptic curve [i.e., a hyperbolic curve of type is a F non .Write E that admits stable reduction over all v ∈ V ( F ) (1 , 1)] over F F for the F determined by X elliptic curve [so X over ⊆ E ]; F F F X C → F F for the § 0] over F obtained by forming the stack- hyperbolic orbicurve [cf. by the unique F -involution [i.e., automorphism X theoretic quotient of F [cf., e.g., field of moduli for the F ; F ⊆ of order two] “ 1” of − X F mod [AbsTopIII], Definition 5.1, (ii)] of X F ⊆ F for the maximal solvable ; F sol def in F ; V = V ( F ). Then of F extension mod mod mod bad V ⊆ V mod mod nonempty set F is a of odd residue of nonarchimedean valuations of mod has bad [i.e., multiplicative] reduction at the X characteristic such that F def good bad bad ⊆ V .Write V = V V \ ( V V ) that lie over F elements of mod mod mod mod mod X bad may in fact have [where we note that reduction at some of the F def good   ) !]; V ( V ) ⊆ V = V F × ( F ) that lie over V V elements of ( F mod V mod mod mod  ∈{ bad , good } ; for def def ( ) = π π ( X = ) ⊆ Π C Π C 1 1 X F F F F def def = π ) ( X F × × F ) C Δ ( ⊆ = π Δ 1 1 F F F F C X for the ́ etale fundamental groups [relative to appropriate choices of base- points] of X natural exact , C .[Thus,wehave , X F × × F , C F F F F F F → G → − Π 1 for “( )” taken to be either → Δ → 1 sequences F − ( ( − ) ) F “ X ”or“ C ”.] Here, we suppose further that the field extension F/F mod

62 62 SHINICHI MOCHIZUKI Galois of degree l , and that the 2 · 3 -torsion points of E is are prime to F over F rational . prime number l (c) ≥ 5 such that the image of the outer homomorphism is a G ( F → GL ) l 2 F contains the subgroup SL ( F ⊆ ) l -torsion points determined by the E of l 2 F GL F ); write ( K ⊆ F for the finite Galois extension of F determined by l 2 is l prime the kernel of this homomorphism. Also, we suppose that to the bad orders ,aswellastothe V [residue characteristics of the] elements of mod [i.e., in the terminology of [GenEll], Definition of -parameters E q of the F bad ] at the primes of ( F ) . V “local heights” of E 3.3, the F C (d) hyperbolic orbicurve of type (1 -tors) [cf. [EtTh], Definition 2.1] is a ,l ± K K over K -core [cf. [CanLift], Remark 2.1.1; [EtTh], the discussion ,with def = C . [Thus, by (c), it follows × K C § at the beginning of 2] given by F K F is completely determined , up to isomorphism over F ,by C .] In C that F K determines, up to -isomorphism, a hyperbolic orbicurve K C particular, K of type (1 ,l -tors) [cf. [EtTh], Definition 2.1] over K , together with X K natural cartesian diagrams −→ Δ −→ −→ X Δ Π Π X X F X X X K F K ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       C −→ −→ Π Δ −→ Π C Δ F C C C C K F K of finite ́ etale coverings of hyperbolic orbicurves and corresponding open immersions of profinite groups. Finally, we recall from [EtTh], Proposition ⊆ Δ admits uniquely determined open subgroups Δ ⊆ 2.2, that Δ X C C Δ , which may be thought of as corresponding to finite ́ etale coverings C def Θ = C × , F by hyperbolic orbicurves ) -tors X C ,l of type (1 , C of F F F F Θ -tors ,l (1 ) , respectively [cf. [EtTh], Definition 2.3]. ± (e) V V ( K ) is a subset that induces a natural bijection ⊆ ∼ V V → mod def non V ( K )  V — i.e., a .Write V section of the natural surjection = mod ⋂ ⋂ ⋂ def def def bad arc good good non arc V K K ) ( V ( ( K ) ) V V , V = , = , V V V = V ⋂ bad V ( K ) .Foreach v ∈ V ( K ), we shall use the subscript v to de- V note the result of base-changing hyperbolic orbicurves over F or K to natural ), we have . Thus, for each v ∈ V ( K ) lying under a v ∈ V ( F K v cartesian diagrams Π −→ Π −→ Δ −→ X −→ X X X X X v v v v v ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       C C −→ −→ C Δ −→ Π Π −→ v C C C v v v v of profinite ́ etale coverings of hyperbolic orbicurves and corresponding injections of profinite groups [i.e., ́ etale fundamental groups ]. Here, the

63 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 63 denotes base-change with respect to F↪ F v ; the various subscript → v ”admit natural outer surjections onto the decom- profinite groups “Π − ) ( def G ) determined, up to ⊆ = Gal( F/K -conjugacy, G position group G K K v bad ∈ V .If , then we assume further that the hyperbolic orbicurve v by v is of type (1 , [cf. [EtTh], Definition 2.5, (i)]. [Here, we note /l Z ) Z C ± v -torsion points that the that it follows from the portion of (b) concerning 2 ̈ satisfies the assumption “ K K ” of [EtTh], Definition 2.5, = base field K v bad V , it follows from the theory ∈ v (i).] Finally, we observe that when 2 — i.e., roughly speaking, “by extracting an l -th root of the of [EtTh], § , C admit natural models X theta function” —that v v X ,C v v Θ Θ ( (1 , ( Z /l Z ) ), (1 over , , which are hyperbolic orbicurves Z /l Z ) of type ) K , ± v respectively [cf. [EtTh], Definition 2.5, (i)]; these models determine open bad V ⊆ Π , then, relative to the notation ∈ ⊆ Π v .If subgroups Π C C X v v v def tp of Remark 3.1.1 below, we shall write Π =Π . v X v is a cusp of the hyperbolic orbicurve C [cf. (d)] that arises from (f)  K nonzero element of the quotient “ Q ” that appears in the definition of a ” given in [EtTh], Definition ,l a “hyperbolic orbicurve of type (1 -tors) ± determined by ∈ V , then let us write  C for the cusp of v 2.1. If v v bad .If v ∈ V is the cusp that arises from the , then we assume that   v ̂ ± Z ” that appears [up to sign] “ canonical generator 1” of the quotient “ ” given in Z ) in the definition of a “hyperbolic orbicurve of type (1 , Z /l ± def , = X K, C × ) [EtTh], Definition 2.5, (i). Thus, the data ( X K F F K hyperbolic orbicurves determines ,C X − → − → K K ,l (1 of type ) , (1 -tors -tors ) , respectively [cf. Definition 1.1, Remark ,l ± − → → − − − ⊆ ⊆ , ⊆ Π Δ Δ ⊆ Π ,Δ 1.1.2], as well as open subgroups Π X C C C C X F → − → − − → − → K K good good ∈ , then we shall write V ,Π V ∈ v .If ⊆ Π Π ⊆ v and, for C X C v → → − − v v def Π =Π . X v − → v Relative to the notation of Definition 3.1, (e), suppose that Remark 3.1.1. non v ∈ . Then in addition to the various profinite groups Π ,Δ V ,onealso ( ( − − ) ) v has corresponding tempered fundamental groups tp tp ;Δ Π ) ( ( − ) − v v [cf. [Andr ́ e], § 4; [SemiAnbd], Example 3.10], whose profinite completions may be ”, the topological . Here, we note that unlike “Δ ,Δ identified with Π ( − ) ( − ) − ) ( v tp group Δ depends, a priori, on v . ( − ) v

64 64 SHINICHI MOCHIZUKI Remark 3.1.2. (i) Observe that the open subgroup Π may be constructed group- ⊆ Π C X K K . Indeed, it follows immediately from theoretically from the topological group Π C K C ” in the discussion at the beginning ”, “ the construction of the coverings “ X § of [EtTh], 2 [cf. also [AbsAnab], Lemma 1.1.4, (i)], that the closed subgroup ⊆ Π may be characterized by a rather simple explicit algorithm. Since the Δ X C K — i.e., the cusps whose inertia at the nonzero cusps decomposition groups of Π C K [cf. the discussion at the beginning of § 1] — are also groups are contained in Δ X [cf., e.g., [AbsTopI], Lemma 4.5, as well as Remark 1.2.2, (ii), of the group-theoretic present paper], the above observation follows immediately from the easily verified fact that the image of any of these decomposition groups associated to nonzero . in Π Δ / cusps coincides with the image of Π X X C K K (ii) In light of the observation of (i), it makes sense to adopt the following convention: of [Ab- Instead of applying the group-theoretic reconstruction algorithm sTopIII], Theorem 1.9 [cf. also the discussion of [AbsTopIII], Remark ], we [or topological groups isomorphic to Π 2.8.3], directly to Π C C K K ⊆ Π shall apply this reconstruction algorithm to the open subgroup X K to reconstruct the function field of , equipped with its natural X Π C K K ∼ Π -action ) . Π /C / Gal( X = X C K K K K In this context, we shall refer to this approach of applying [AbsTopIII], Theorem 1.9, ⋂ non good V V ∈ -approach as the Θ v to [AbsTopIII], Theorem 1.9. Note that, for bad ∈ ), one may also adopt a “Θ-approach” to applying [Ab- v (respectively, V ,Π or [by applying Corollary 1.2] Π (respec- sTopIII], Theorem 1.9, to Π X C C v → − → − v v tp tp ). In the present series or [by applying [EtTh], Proposition 2.4] Π tively, to Π X C v v of papers, we shall always think of [AbsTopIII], Theorem 1.9 [as well as the other results of [AbsTopIII] that arise as consequences of [AbsTopIII], Theorem 1.9] as be- ⋂ good non bad V ∈ V v V or, for ), (respectively, v ∈ ing applied to [isomorphs of] Π C K tp tp Π (respectively, Π ) via the “Θ-approach” [cf. also Remark ,Π ,Π ,Π C X C X C v − → → − v v v v 3.4.3, (i), below]. (iii) Recall from the discussion at the beginning of [EtTh], 2, the tautological § extension ell Θ 1 → Δ Δ → → 1 → Δ Θ X X def def def ell Θ , Δ ]]; Δ ] / [Δ Δ , [Δ , , Δ [Δ ]]; Δ = , =[Δ =Δ [Δ / —whereΔ X X X X X X X X X Θ X X ab 2 ell ∈ H tautological (Δ . The extension class ) of this extension determines a Δ , Δ Θ X X isomorphism ∼ → Δ M Θ X — where we recall from [AbsTopIII], Theorem 1.9, (b), that the module “ ” M X of [AbsTopIII], Theorem 1.9, (b) [cf. also [AbsTopIII], Proposition 1.4, (ii); [Ab- 2 ell ̂ ̂ ). (Δ Z ) , , Z with Hom( naturally identified sTopIII], Remark 1.10.1, (ii)], may be H X In particular, we obtain a tautological isomorphism ∼ ) Δ → ( l · M Θ X

65 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 65 :Δ l ]. In particular, we observe that if we write Π [i.e., since [Δ ]= X X C F mod etale fundamental group of the orbicurve discussed in Remark 3.1.7, C for the ́ F mod ∼ (i), below, then M → · Δ l ) may be regarded as a characteristic subquotient ( Θ X .Fromthepoint , hence admits a natural conjugation action by Π of Π C C F F mod mod of the “ Θ significance of view of the theory of the present series of papers, the - ∼ → · ( l M lies precisely in the existence of this tautological isomorphism approach” X bad v . That is to say, the Θ-approach V ), which will be applied in [IUTchII] at ∈ Δ Θ involves applying the reconstruction algorithm of [AbsTopIII], Theorem 1.9, via the , which may be identified, via the above tautological isomorphism, M cyclotome X ), which plays a central role in the theory of [EtTh] — l · Δ with the cyclotome ( Θ “cyclotomic rigidity” cf., especially, the discussion of in [EtTh], Corollary 2.19, (i). “large” as being (iv) If one thinks of the prime number l , then the role played “ Θ in the above discussion of the is reminiscent of the -approach” X by the covering role played by the universal covering of a complex elliptic curve by the complex plane 2 [cf., e.g., [AbsTopIII], in the holomorphic reconstruction theory of [AbsTopIII], § Propositions 2.5, 2.6]. bad ∅  = [cf. Definition 3.1, (b)], it follows immediately V Since Remark 3.1.3. mod bad from Definition 3.1, (d), (e), (f), that the data ( ,l,C )is, , F/F, X , V V , F K mod bad in fact, completely determined by the data ( ,C F/F, X , V , V ), and that F K mod C is completely determined up to K -isomorphism by the data ( F/F, X ). ,l, V F K bad V ), distinct choices of “ ,l, V ” will Finally, we remark that for given data ( X F mod not affect the theory in any significant way. bad It follows immediately from the definitions that at each v Remark 3.1.4. V ∈ [which is necessarily — cf. Definition 3.1, (c)] (respectively, each v ∈ l prime to ⋂ ⋂ good non good non V V v ∈ V ; which is prime to l ), X ; each (respectively, X V − → v v X stable model . K )admitsa over the ring of integers of v v Remark 3.1.5. -torsion points of E Notethatsincethe3 are rational over , F F [cf. Definition 3.1, (b)], it follows [cf., e.g., [IUTchIV], F Galois over F and is mod . In addition to working with is Galois over F Proposition 1.8, (iv)] that K mod the field F and various extensions of contained in F , we shall also have F mod mod algebraic stack occasion to work with the def ) = Spec( O K/F ) // Gal( S K mod mod stack-theoretic quotient [i.e., “ // ”] of the spectrum of the obtained by forming the of K by the Galois group Gal( K/F ). Thus, any finite exten- O ring of integers K mod sion L ⊆ F of F ,an in F determines, by forming the integral closure of S L in mod mod arithmetic line bun- . In particular, by considering S over algebraic stack S ,L mod mod ) Q  , one may associate to any finite quotient Gal( F/F over such dles S mod ,L mod a Frobenioid via [the easily verified “stack-theoretic version” of] the construction of [FrdI], Example 6.3. One verifies immediately that an appropriate analogue of [FrdI], Theorem 6.4, holds for such stack-theoretic versions of the Frobenioids con- structed in [FrdI], Example 6.3. Also, we observe that upon passing to either the

66 66 SHINICHI MOCHIZUKI or the realification naturally iso- perfection , such stack-theoretic versions become morphic to the non-stack-theoretic versions [i.e., of [FrdI], Example 6.3, as stated]. In light of the important role played by the various orbicurves Remark 3.1.6. 2, in the present series of papers, we take the opportunity § constructed in [EtTh], to correct an unfortunate — albeit in fact ! — error in [EtTh]. In the irrelevant discussion preceding [EtTh], Definition 2.1, one must in fact assume that the integer Δ to be well-defined . Since, ultimately, in [EtTh] in order for the quotient l is odd X [cf. the discussion following [EtTh], Remark 5.7.1], as well as in the present series of papers, this is the only case that is of interest, this oversight does not affect either the present series of papers or the bulk of the remainder of [EtTh]. Indeed, l is used are [EtTh], Remark 2.2.1, the only places in [EtTh] where the case of even and the application of [EtTh], Remark 2.2.1, in the proof of [EtTh], Proposition ̇ ”. Thus, [EtTh], Remark 2.2.1, must be ;in deleted 2.12, for the orbicurves “ C [EtTh], Proposition 2.12, one must in fact exclude the case where the orbicurve ̇ ”. On the other hand, this theory involving [EtTh], C under consideration is “ Proposition 2.12 [cf., especially, [EtTh], Corollaries 2.18, 2.19] is only applied after the discussion following [EtTh], Remark 5.7.1, i.e., which only treats the curves ”. That is to say, ultimately, in [EtTh], as well as in the present series of papers, X “ ”, whose treatment only requires the case of one is only interested in the curves “ X odd . l Remark 3.1.7. K - and the F (i) Observe that it follows immediately from the definition of mod [up to unique [cf. Definition 3.1, (b), (d)] that C admits a unique C of coricity K F isomorphism] model C F mod F for the result of base-changing this .If v over V C , then we shall write ∈ mod mod v . When applying the group-theoretic reconstruction algorithm ) F model to ( v mod of [AbsTopIII], Theorem 1.9 [cf. Remark 3.1.2, (ii)], it will frequently be useful to , as follows. C and special types of rational functions consider certain C on v F mod . V ∈ v for some ) F or to ( which is equal either to field be a L Let F mod mod mod v C over L . Thus, one verifies immediately for the model just discussed of Write C L F is isomorphic to C associated to the algebraic stack | coarse space | that the C L L over affine line . Now suppose that we are given an algebraic closure the L L C function field L of the of C determined .Write L for the algebraic closure of L L C L . We shall refer to a closed point of the proper smooth curve determined by C if it maps to a closed L critical point of L as a ⊆ by some finite subextension C C cpt | that arises from one | C of | C | point of the [proper smooth] compactification L L ; we shall refer to a critical point which does not of the 2 -torsion points of E F cpt | strictly that arises from the unique cusp as C of C map to the closed point of | L L . Thus, as one might imagine from the 2 -torsion points central importance of critical cpt may | C , the strictly critical points of in the elementary theory of elliptic curves | L be thought of as the of “most fundamental/canonical non-cuspidal points” cpt | —wherewe . We shall refer to a rational function f -coric L κ on C as ∈ C | C L L think of the κ as standing for “Kummer” —if

67 ̈ INTER-UNIVERSAL TEICHM 67 ULLER THEORY I has  , it holds that, over L , f L precisely one pole [of · whenever ∈ f unrestricted order], but at least two distinct zeroes ; of the divisor of is defined over a number field and · f zeroes and poles ; critical points avoids the cpt | . every strictly critical f C · at root of unity restricts to a point of | L Thus, the first displayed condition, taken together with the latter portion of the there exist second displayed condition, may be understood as the condition that cpt | with respect to which [i.e., if one a unique non-critical L -rational point of | C L -rational point to be the “point at infinity”] f may be thought of as a takes this L affine line L with non-critical zeroes . In particular, it over on the polynomial  ∈ L ,itis never the case follows from the first displayed condition that, whenever f − 1 κ are . By contrast, the third displayed condition may be -coric f that both and f points determines understood as the condition that restriction to the strictly critical important canonical splitting up to roots of unity [which will play an a sort of in the present series of papers — cf., e.g., the discussion of Example 5.1, (v); role Definition 5.2, (vi), (viii); Remark 5.2.3, below] of the set of nonzero constant [i.e., -coric functions into a direct product, up to roots of unity, of L κ -] multiples of the set of L . In particular, it κ -coric functions and the set of nonzero elements of are such that L and f ∈ L follows from the third displayed condition that if c ∈ C f and c · f are κ -coric ,then c is a root of unity . both  intermediate field between be an (ii) We maintain the notation of (i). Let L that is solvably closed [cf. [GlSol], Definition 1, (i)], i.e., has no nontrivial L and L cpt critical points 4 | precisely has ,it abelian extensions. Observe that, since | C L elementary theory of polynomial functions on follows immediately from the cpt | of some L -rational point L [i.e., the complement in | C the affine line over L cpt L ∈ | [i.e., a rational function on the affine ] that there exists a κ -coric f C | sol L C ] of degree 4. In particular, it follows immediately L from the elementary line over | ] over [together with L theory of polynomial functions on the affine line [i.e., C | L “Hensel’s lemma” — cf., e.g., the method of proof of [AbsTopII], Lemma 2.1] [together with the well-known fact that the f (respectively, from the existence of sol letters is solvable ]) that 4 symmetric group on  -coric ) appears as a value of some κ (respectively, L every element of L  C rational function on L - (respectively, L C -) valued point of at some L L . that is not critical × L ;if ,thenwrite U L for the group L = of nonzero elements of F = If L mod L for the group of units [i.e., relative ) U v ∈ V ,thenwrite for some F ( v mod mod L to the unique valuation on that extends v ]of L . We shall say that an element L n ∈ f -coric is κ κ -coric if there exists a positive integer n such that f L is a ∞ C is ; we shall say that an element f ∈ L if there exists -coric × κ L element of ∞ C C is L ∈ such that c · f ∈ L f is . Thus, an element κ -coric ∈U c an element ∞ C C L . Also, one verifies immediately that κ -coric κ -coric if and only if it is ∞ an κ L -coric element f ∈ if and only if it restricts to a × is -coric κ ∞ C ∞ root of unity at some [or, equivalently, every ] strictly critical point of the L of the proper smooth curve determined by some finite subextension ⊆ C . f that contains L function field C

68 68 SHINICHI MOCHIZUKI Finally, one verifies immediately that the operation of multiplication determines a κ -, and κ × -coric rational [cf. κ structure of -, § pseudo-monoid 0] on the sets of ∞ ∞ κ κ × -coric rational functions, the re- -and functions; moreover, in the case of ∞ ∞ . These pseudo-monoids will be cyclotomic and sulting pseudo-monoid is divisible C of use in discussions concerning the Kummer theory of rational functions on L [cf. Example 5.1, (i), (v); Definition 5.2, (v), (vi), (vii), (viii), below]. , L (iii) We maintain the notation of (i) and (ii) and assume further that F = mod × -solvable ∈ L = is κ . We shall say that an element if it is an F F f -multiple L C sol [cf. Definition 3.1, (b)] of a -coric element of . Thus, one verifies immediately κ L C ∞ ∈ that an element f is κ -solvable if and only if there exists a positive integer L C n ( is a ⊆ κ × -coric element of F ) · L for the .Write F K μ f such that n l ∞ C sol generated by the -th roots of unity; L K l subextension of κ -sol) ⊆ L ( for the C C subfield of elements of generated by the κ -solvable L L for the ; L L ( C ⊆ ) C C C C K subfield of · F ( μ ) by the images of the L -linear embeddings L generated over L C C l C into of the of C L function field F/F is . Thus, the fact that the extension mod C K prime to [cf. Definition 3.1, (b)] implies that Galois l of degree the subgroup Gal( K/F ( μ ⊆ Gal( K/F )) )is normal and may be char- mod l )thatis[abstractly] of Gal( K/F acterized as the unique subgroup mod ( F ) isomorphic to SL 2 l [cf. Remark 3.1.5; [GenEll], Lemma 3.1, (i)]. Moreover, we observe that it follows SL immediately from the well-known fact that the finite group ( F )is perfect [cf. 2 l Definition 3.1, (c); [GenEll], Lemma 3.1, (ii)], together with the definition of the zeroes and poles avoid the critical [cf., especially, the fact that the κ × -coric” term “ ∞ points !], that κ ( ( C linearly disjoint ) ⊆ L -sol) are ⊇ F ( μ L ) · L the subfields l C C C K ) · L . over μ ( F C l natural isomorphism In particular, it follows that there is a ∼ Gal( L C κ ) /F ( μ ( ) · L L ) ( → Gal( L · ( C ) ) · L μ ( κ -sol) /F ( -sol)) C l C C C l C K K μ ( C action ) /F ( ) as being equipped with an L ) · L — i.e., one may regard Gal( l C C K -sol). κ ( C ( ) · L L ( κ -sol) that restricts to the trivial action on F ( μ · ) L on l C C C K (iv) We maintain the notation of (iii). In the following, we shall write “Out( − )” for the group of outer automorphisms of the topological group in parentheses. Con- sider the tautological exact sequence of Galois groups L κ 1 /L → ( κ -sol)) → Gal( L ) /L /L ) → Gal( L ( -sol) → Gal( 1 C C C C C C -sol)) as L /L κ ( [cf. the discussion of (iii)]. Let us refer to a subgroup of Gal( C C a κ -sol -open subgroup if it is the intersection with Gal( L -sol)) of a normal /L κ ( C C open subgroup of Gal( L /L ). Thus, the subgroups C C κ -sol (Gal( L -sol))) ( κ -sol))) ⊆ Aut(Gal( L /L κ ( /L Aut C C C C -sol κ Out (Gal( L κ /L ( ( κ -sol))) ⊆ Out(Gal( L /L -sol))) C C C C

69 ̈ INTER-UNIVERSAL TEICHM 69 ULLER THEORY I L of automorphisms/outer automorphisms of the topological group Gal( /L ( κ -sol)) C C -outer κ -sol -automorphisms/ κ -sol -sol-open subgroup — i.e., of that preserve each “ κ natural compatible homomorphisms —admit automorphisms” κ -sol L ) /L Q ( (Gal( -sol))) → Aut( κ Aut C C -sol κ Out κ /L L ( (Gal( -sol))) → Out( Q ) C C L κ /L  ( κ -sol)) -sol-open subgroup. The kernels of Q by a for each quotient Gal( C C Q ”] determine these natural homomorphisms [for varying “ natural profinite topolo- -sol κ -sol κ L -sol))), Out -sol))), with respect to ( κ κ L (Gal( (Gal( /L ( /L on Aut gies C C C C commutative diagram of homomorphisms whicheacharrowofthe κ -sol Gal( -sol))) ) −→ Aut L /L (Gal( L κ /L ( C C C C ⏐ ⏐ ⏐ ⏐   κ -sol Gal( L κ -sol) /L κ ) −→ Out ( -sol))) (Gal( L ( /L C C C C exact sequence considered above is . that arises, via conjugation, from the continuous Finally, we observe that Gal( center-free L ( /L -sol)) is ; in particular, the above commutative κ C C diagram of homomorphisms of topological groups is cartesian . Indeed, let us first observe that it follows immediately from the definitions that F · L abelian ( κ -sol) / F · L . Thus, it follows formally, by applying Lemma )is Gal( C C geometric fundamental groups of the various 2.7, (vi), (vii), to the genus zero affine by conjugacy action , that the hyperbolic curves is equal to function field whose L C L -sol)) on such a [center-free] geometric /L κ ( any element center of Gal( α in the C C fundamental group is trivial and hence, by [the special case that was already known to Belyi of] the Galois injectivity result discussed in [NodNon], Theorem C, that α L /L -sol)), as desired. ( κ is the identity element of Gal( C C initial Θ -data as in Definition 3.1, the theory of Frobenioids given in Given associated Frobenioids ,as [FrdI], [FrdII], [EtTh] allows one to construct various follows. ∈ Example 3.2. Frobenioids at Bad Nonarchimedean Primes. Let v ⋂ bad bad ( K ) V V dis- “Frobenioid-theoretic theta function” . The theory of the = V cussed in [EtTh], 5, may be thought of as a sort of formal, category-theoretic way § elementary classical facts [which are reviewed in [EtTh], § 1] to formulate various concerning the theory of the line bundles and divisors related to the classical theta function Tate curve over an MLF. We give a brief review of this theory of on a [EtTh], § 5, as follows: tempered determines a (i) By the theory of [EtTh], the hyperbolic curve X v Frobenioid F v

70 70 SHINICHI MOCHIZUKI § 5; ” in the discussion at the beginning of [EtTh], C [i.e., the Frobenioid denoted “ cf. also the discussion of Remark 3.2.4 below] over a base category D v [i.e., the category denoted “ D § 5]. ” in the discussion at the beginning of [EtTh], may be thought of as the category D We recall from the theory of [EtTh] that v temp 0 — i.e., “ B of connected tempered coverings ( X ) ” in the notation of [EtTh], v Example 3.9 — of the hyperbolic curve X . In the following, we shall write v def  0 B ( K = ) D v v [cf. the notational conventions concerning categories discussed in § 0]. Also, we  etale coverings may be naturally regarded [by pulling back finite ́ D observe that v full subcategory Spec( K )] as a → via the structure morphism X v v  ⊆D D v v  left-adjoint , and that we have a natural functor D to →D of D ,whichis v v v  ) ↪ →D − [cf. [FrdII], Example 1.3, (ii)]. If ( the natural inclusion functor D v v is an object of D T ”the [a Frobenius-trivial object , then we shall denote by “ v ) − ( notion which is category-theoretic — cf. [FrdI], Definition 1.2, (iv); [FrdI], Corollary [which is completely determined up to F 4.11, (iv); [EtTh], Proposition 5.1] of v isomorphism] that lies over “( )”. − (ii) Next, let us recall [cf. [EtTh], Proposition 5.1; [FrdI], Corollary 4.10] that the birationalization def birat ÷ = F F v v may be reconstructed category-theoretically from [cf. Remark 3.2.1 below]. Write F v ̈ → X Y v v log ̈ for the tempered covering determined by the object “ ” in the discussion at the Y ̈ § Y 5. Thus, we may think of beginning of [EtTh], as an object of D [cf. the v v bs object “ A ”of[EtTh], § 5, in the “double underline case”]. Then let us recall the  “Frobenioid-theoretic l -th root of the theta function” ,whichis normalized so as to √ 1” [cf. [EtTh], Theorem 5.7]; we shall denote − attain the value 1 at the point “ the reciprocal of [i.e., “1 over”] this theta function by ÷ × ∈O ( T ) Θ ̈ v Y v ÷ F of an object of . F ” to denote the image in ÷ — where we use the superscript “ v v Here, we recall that Θ -th root is completely determined up to multiplication by a 2 l v ÷ )] and the action of the group of automorphisms ( T of unity μ [i.e., an element of l 2 ̈ Y v

71 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 71 Z · T ⊆ l Aut( ) [i.e., we write Z for the group denoted “ Z ” in [EtTh], Theorem ̈ Y v 5.7; cf. also the discussion preceding [EtTh], Definition 1.9]. Moreover, we recall § 5; § 5 [cf. the discussion at the beginning of [EtTh], from the theory of [EtTh], [EtTh], Theorem 5.7] that [regarded up to isomorphism] and T ̈ Y v ÷ μ Z · ( T l indeterminacies discussed above] ), [regardeduptothe Θ l 2 ̈ v Y v F may be reconstructed category-theoretically from [cf. Remark 3.2.1 below]. v (iii) Next, we recall from [EtTh], Corollary 3.8, (ii) [cf. also [EtTh], Proposition p 5.1], that the -adic Frobenioid “base-field-theoretic hull” [cf. constituted by the v [EtTh], Remark 3.6.2] ⊆F C v v bs-fld for the subcategory “ C [i.e., we write ” of [EtTh], Definition 3.6, (iv)] may C v [cf. Remark 3.2.1 below]. be F reconstructed category-theoretically from v (iv) Write q for the q -parameter of the elliptic curve E .Thus,wemay over K v v v   ∼ think of q ( T ). Note that it follows from our as- ∈O )( q as an element O = X v v K v v [cf. Definition 3.1, (b)], together with the definition sumption concerning 2 -torsion   ∼ ” [cf. Definition 3.1, (c)], that q K of “ O ( admits a 2 l -th root )( in O ). T = X v K v v Then one computes immediately from the final formula of [EtTh], Proposition 1.4, √ (ii), that the Θ value of at q is equal to − v v def 1 / 2 l  q q = ∈O ) ( T X v v v / l 1 2 q — where the notation “ q ]is completely determined up to a ” [hence also v v μ ( T [cf. [FrdI], Definition 1.1, divisor monoid ) -multiple .WriteΦ for the 2 X l C v v p (iv)] of the q C . Then the image of -adic Frobenioid determines a constant v v v isomorphic to N ]“log .Moreover, ( q )” of Φ D section [i.e., a sub-monoid on C v Φ v v the resulting submonoid [cf. Remark 3.2.2 below] def ⊆ | ( = N · log ) | q Φ Φ    C C D D Φ v v v v v  [cf. [FrdII], Example base category given by D determines a -adic Frobenioid with p v v 1.1, (ii)] ÷  C ⊆F ⊆C →F ( ) v v v v — which may be thought of as a subcategory of C . Also, we observe that [since the v of -orbit characteristic ∈ K , it follows that] q ) determines a μ − ( -parameter q q v l 2 v v splittings [cf. [FrdI], Definition 2.3]  τ v

72 72 SHINICHI MOCHIZUKI  C . on v ̈ is equal to K [cf. the discussion of base field (v) Next, let us recall that the Y v v of Definition 3.1, (e)]. Write Θ D ⊆ ( D ) ̈ v v Y v full subcategory of the category ( D ) for the [cf. the notational conventions ̈ v Y v ̈ products in D § concerning categories discussed in 0] determined by the of Y v v  with objects of . Thus, one verifies immediately that “forming the product D v ∼ Θ  ̈ ” determines a . Moreover, for D natural equivalence of categories →D Y with v v v Θ Θ A D ), the assignment ∈ Ob( v N ÷ × × Θ Θ | ) ·  ( ) T →O T ( ) ⊆O (Θ A T Θ A Θ A A v  Θ determines a O monoid ) ( − [in the sense of [FrdI], Definition 1.1, (ii)]; on D Θ v C v ×  write O ( ( − ) ⊆O ) for the submonoid determined by the invertible elements. − Θ Θ C C v v ∼  Θ →D D Next, let us observe that, relative to the natural equivalence of categories v v def  Θ Θ ̈  A  → A — which we think of as mapping Ob( D = ∈ Y )—we × A ) Ob( D v v v have natural isomorphisms ∼ ∼ × ×   O →O ) − ( ( ( − ); O − →O ) ) ( − Θ  Θ  C C C C v v v v ×   ( − ), as in ( C ) are the monoids associated to the Frobenioid − O O [where   v C C v v [FrdI], Proposition 2.2] which are compatible with the assignment q |  → Θ | T T Θ A A v v Θ ̈ A = Y and the natural isomorphism [i.e., induced by the natural projection × v ∼  × × Θ T T ) ) − ( →O ( ( A A → ] O ). In particular, we conclude that the monoid O A Θ A C v determines — in a fashion consistent with the notation of [FrdI], Proposition 2.2! Θ -adic Frobenioid base category given by D with [cf. [FrdII], Example 1.1, p —a v v (ii)] ÷ Θ ( ) ⊆F C v v ÷ F — which may be thought of as a subcategory of , and which is equipped with a v [cf. [FrdI], Definition 2.3] characteristic splittings ( − ) -orbit of μ l 2 Θ τ v determined by Θ natural equivalence of categories . Moreover, we have a v ∼  Θ C →C v v

73 ̈ INTER-UNIVERSAL TEICHM 73 ULLER THEORY I Θ  to . This fact may be stated more succinctly by writing τ that maps τ v v ∼ Θ  F →F v v def def    Θ Θ Θ =( ). In the following, we shall refer ); F C —wherewewrite F =( C ,τ ,τ v v v v v v  Θ toapairsuchas F F or consisting of a Frobenioid equipped with a collection v v . of characteristic splittings as a split Frobenioid (vi) Here, it is useful to recall [cf. Remark 3.2.1 below] that:  may be ⊆D reconstructed category-theoretically (a) the subcategory D v v [cf. [AbsAnab], Lemma 1.3.8]; from D v Θ (b) the category D may be reconstructed category-theoretically from D [cf. v v 5]; § (a); the discussion at the beginning of [EtTh],  Θ )maybe (respectively, reconstructed category-theoretically D (c) the category D v v  Θ C from (respectively, C ) [cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem v v 1.2, (i); [FrdII], Example 1.3, (i); [AbsAnab], Theorem 1.1.1, (ii)]; (d) the category D may be reconstructed category-theoretically either from v [cf. [EtTh], Theorem 4.4; [EtTh], Proposition 5.1] C [cf. or from F v v [FrdI], Theorem 3.4, (v); [FrdII], Theorem 1.2, (i); [FrdII], Example 1.3, (i); [SemiAnbd], Example 3.10; [SemiAnbd], Remark 3.4.1]. Next, let us observe that by (b), (d), together with the discussion of (ii) concerning category-theoreticity of Θ the , it follows [cf. Remark 3.2.1 below] that v Θ reconstruct the split Frobenioid F (e) one may [up to the · Z indeterminacy l v category-theoretically discussed in (ii); cf. also Remark 3.2.3 below] in Θ v F from [cf. [FrdI], Theorem 3.4, (i), (v); [EtTh], Proposition 5.1]. v values of Θ Next, let us recall that the may be computed by restricting the cor- v responding Kummer class , i.e., the “ ́ etale theta function” [cf. [EtTh], Proposition 1.4, (iii); the proof of [EtTh], Theorem 1.10, (ii); the proof of [EtTh], Theorem 5.7], D from which may be reconstructed category-theoretically [cf. [EtTh], Corollary v of [AbsTopIII], Corol- 2.8, (i)]. Thus, by applying the isomorphisms of cyclotomes lary 1.10, (c); [AbsTopIII], Remark 3.2.1 [cf. also [AbsTopIII], Remark 3.1.1], to these Kummer classes, one concludes from (a), (d) that  category-theoretically from (f) one may reconstruct the split Frobenioid F v , hence also [cf. (iii)] from F [cf. Remark 3.2.1 below]. C v v Remark 3.2.1. (i) In [FrdI], [FrdII], and [EtTh] [cf. [EtTh], Remark 5.1.1], the phrase “recon- structed category-theoretically” is interpreted as meaning “preserved by equivalences of categories” . From the point of view of the theory of [AbsTopIII] — i.e., the dis- cussion of “mono-anabelian” versus “bi-anabelian” geometry [cf. [AbsTopIII], § I2,

74 74 SHINICHI MOCHIZUKI (Q2)] — this sort of definition is “bi-anabelian” in nature. In fact, it is not difficult explicit re- to verify that the techniques of [FrdI], [FrdII], and [EtTh] all result in consists solely of the category structure ,whose construction algorithms input data “mono-anabelian” nature that do not require the use of of the given category, of a [cf. the discussion of some fixed reference model that arises from scheme theory foundational aspects of such “mono-anabelian I4]. For more on the § [AbsTopIII], reconstruction algorithms”, we refer to the discussion of [IUTchIV], Example 3.5. (ii) One reason that we do not develop in detail here a “mono-anabelian ap- proach to the geometry of categories” along the lines of [AbsTopIII] is that, unlike quite essen- the case with the mono-anabelian theory of [AbsTopIII], which plays a tial role in the theory of the present series of papers, much of the category-theoretic reconstruction theory of [FrdI], [FrdII], and [EtTh] is not of essential importance in the development of the theory of the present series of papers. That is to say, for instance, instead of quoting results to the effect that the base categories or divisor monoids of various Frobenioids may be reconstructed category-theoretically, one could instead simply work with the data consisting of “the category constituted by the Frobenioid equipped with its pre-Frobenioid structure” [cf. [FrdI], Definition 1.1, (iv)]. Nevertheless, we chose to apply the theory of [FrdI], [FrdII], and [EtTh] simplifies the exposition [i.e., reduces the number of auxiliary partly because it structures that one must carry around], but more importantly because it renders explicit precisely which structures arising from scheme-theory are “categorically and which merely amount to “arbitrary, non-intrinsic choices” which, intrinsic” “indeterminacies” when formulated intrinsically, correspond to various .Thisex- plicitness is of particular importance with respect to phenomena related to the unit- linear Frobenius functor Frobenioid-theoretic [cf. [FrdI], Proposition 2.5] and the § 5. indeterminacies studied in [EtTh], is not “absolutely primitive” in the Remark 3.2.2. Although the submonoid Φ  C v sense of [FrdII], Example 1.1, (ii), it is “very close to being absolutely primitive”, N such that in the sense that [as is easily verified] there exists a positive integer is absolutely primitive . This proximity to absolute primitiveness may also · Φ N  C v  τ be seen in the existence of the characteristic splittings . v Remark 3.2.3. ̈ α ). Then observe that Y determines, in a natural way, an ( Aut ∈ α (i) Let D v v  automorphism α obtained by composing the equivalence of the functor →D D D v v ∼ Θ  Θ  Θ Ob( [i.e., which maps Ob( D D ∈ →D  A  → A )] discussed ) D of categories v v v v Θ D in Example 3.2, (v), with the natural functor .Moreover, ⊆ ( D →D ) ̈ v v v Y v   α − on ) monoid O induces, in a natural way, an isomorphism ( α of the D Θ O C v Θ Θ D associated to Θ in Example 3.2, (v), onto the corresponding monoid on D v v v α α -conjugate Θ . Thus, it follows immediately from the associated to the of Θ v v discussion of Example 3.2, (v), that  α — hence also α — induces an isomorphism of the split Frobenioid O

75 ̈ INTER-UNIVERSAL TEICHM 75 ULLER THEORY I α α Θ Θ split Frobenioid F F associated to Θ associated to Θ onto the which v v v v Θ identity functor on D lies over the . v Θ F “ In particular, the expression , regarded up to the · Z indeterminacy in Θ l v v may be understood as referring to the various split discussed in Example 3.2, (ii)” α Θ ̈ ), relative to the ”, as ( α Y ranges over the elements of Aut F Frobenioids “ D v v v identifications given by these isomorphisms of split Frobenioids induced by the ̈ Y ). ( various elements of Aut D v v Θ ⊆ ) lies in the image of the natural functor D A ∈ (ii) Suppose that D Ob( v v ,andthat is a linear morphism in the Frobenioid F →D ) . ψ : B → T D ( ̈ A v v Y v v induces an injective homomorphism Then ψ ÷ × × ÷ O ) ↪ →O ( ( B T ) A [cf. [FrdI], Proposition 1.11, (iv)]. In particular, one may pull-back sections of the  Θ O monoid of Example 3.2, (v), to ( − ) . Such pull-backs are useful, for D B on Θ v C v 5. § , as in the theory of [EtTh], Θ instance, when one considers the roots of v Before proceeding, we pause to discuss certain minor oversights Remark 3.2.4. on the part of the author in the discussion of the theory of tempered Frobenioids log § be as in the discussion at the beginning of [EtTh], 3. § 4. Let 3, Z in [EtTh], § ∞ log is obtained as the “universal combinatorial covering” of Here, we recall that Z ∞ the formal log scheme associated to a stable log curve with split special fiber over the ring of integers of a finite extension of an MLF of residue characteristic p [cf. log for the generic fiber of the stable log curve Z for more details]; we write loc. cit. under consideration. (i) First, let us consider the following conditions on a nonzero meromorphic log : function Z on f ∞ over some , it holds that f admits an N -th root N (a) For every N ∈ 1 ≥ log . tempered covering of Z -th N admits an f , it holds that p which is prime to N (b) For every N ∈ 1 ≥ log . Z over some tempered covering of root is a (c) The divisor of zeroes and poles of . f log-divisor It is immediate that (a) implies (b). Moreover, one verifies immediately, by consid- ering the ramification divisors of the tempered coverings that arise from extracting f , that (b) implies (c). When N is prime to p ,if f satisfies (c), then roots of admissible coverings [cf., e.g., [PrfGC], it follows immediately from the theory of log log → Z whose pull-back log ́ etale covering Y § 2, § 8] that there exists a finite log log log log Z is sufficient to the generic fiber Z → Z of Y ∞ ∞ ∞ ∞ log that (R1) to annihilate all ramification over the cusps or special fiber of Z ∞ might arise from extracting an N -th root of f ,aswellas

76 76 SHINICHI MOCHIZUKI (R2) to split all extensions of the function fields of irreducible components of log that might arise from extracting an N -th root of the special fiber of Z ∞ . f admits an N -th root over the f That is to say, in this situation, it follows that log log Z tempered covering of given by the “universal combinatorial covering” of Y . In particular, it follows that (c) implies (b). Thus, in summary, we have: ⇒ (b) ⇐⇒ (c) . (a) = On the other hand, unfortunately, it is not clear to the author at the time of writing whether or not (c) [or (b)] implies (a) . § (ii) Observe that it follows from the theory of [EtTh], 1 [cf., especially, [EtTh], Proposition 1.3] that the theta function that forms the main topic of interest of [EtTh] satisfies condition (a) of (i). f as in (i) is defined (iii) In [EtTh], Definition 3.1, (ii), a meromorphic function to be “log-meromorphic” if it satisfies condition (c) of (i). On the other hand, in the proof of [EtTh], Proposition 4.2, (iii), it is necessary to use property (a) of (i) — i.e., despite the fact that, as remarked in (i), it is not clear whether or not property (c) implies property (a). The author apologizes for any confusion caused by this oversight on his part. from the (iv) The problem pointed out in (iii) may be remedied — at least point of view of the theory of [EtTh] — via either of the following two approaches: (A) One may modify [EtTh], Definition 3.1, (ii), by taking the definition of a “log-meromorphic” function to be a function that satisfies condition (a) [i.e., as opposed to condition (c)] of (i). [In light of the content of this modified definition, perhaps a better term for this class of meromorphic functions would be “tempered- meromorphic” .] Then the remainder of the text of [EtTh] goes through without change. (B) One may modify [EtTh], Definition 4.1, (i), by assuming that the meromorphic birat × )” of [EtTh], Definition 4.1, (i), satisfies the following ( A ∈O f function “ “Frobenioid-theoretic version” of condition (a): ′ (d) For every N ∈ N C , there exists a linear morphism A such → A in 1 ≥ ′ N -th root. admits an f to A that the pull-back of [Here, we recall that, as discussed in (ii), the Frobenioid-theoretic theta functions that appear in [EtTh] satisfy (d).] Note that since the rational function monoid of the Frobenioid C , as well as the linear morphisms of C ,are category-theoretic [cf. [FrdI], Theorem 3.4, (iii), (v); [FrdI], Corollary 4.10], this condition (d) is category- theoretic . Thus, if one modifies [EtTh], Definition 4.1, (i), in this way, then the remainder of the text of [EtTh] goes through without change, except that one must replace the reference to the definition of “log-meromorphic” [i.e., [EtTh], Definition 3.1, (ii)] that occurs in the proof of [EtTh], Proposition 4.2, (iii), by a reference to condition (d) [i.e., in the modified version of [EtTh], Definition 4.1, (i)]. (v) In the discussion of (iv), we note that the approach of (A) results in a slightly different definition of the notion of a “tempered Frobenioid” from the original

77 ̈ INTER-UNIVERSAL TEICHM 77 ULLER THEORY I definition given in [EtTh]. Put another way, the approach of (B) has the advantage that it does not result in any modification of the definition of the notion of a “tempered Frobenioid” ; that is to say, the approach of (B) only results in a slight § reduction in the range of applicability of the theory of [EtTh], 4, which is essentially theta from the point of view of the present series of papers, since [cf. (ii)] irrelevant functions lie within this reduced range of applicability. On the other hand, the approach of (A) has the advantage that one may consider the Kummer theory of arbitrary rational functions of the tempered Frobenioid without imposing any . Thus, for the sake of simplicity, in the present series of papers, further hypotheses we shall interpret the notion of a “tempered Frobenioid” via the approach of (A) . ell ” given in [EtTh], (vi) Strictly speaking, the definition of the monoid “Φ W Example 3.9, (iii), leads to certain technical difficulties, which are, in fact, entirely to the theory of [EtTh]. These technical difficulties may be averted by irrelevant making the following slight modifications to the text of [EtTh], Example 3.9, as follows: (1) In the discussion following the first display of [EtTh], Example 3.9, (i), log log is of genus 1” should be replaced by the phrase “ Y is the phrase “ Y genus 1 and has either precisely one cusp or of whose precisely two cusps difference is a 2-torsion element of the underlying elliptic curve”. (2) In the discussion following the first display of [EtTh], Example 3.9, (i), the phrase log log ̇ ̇ X the lower arrow of the diagram to be “ C ” → should be replaced by the phrase log log ̇ ̇ X the lower arrow of the diagram to be “ C → ”. (3) In the discussion following the first display of [EtTh], Example 3.9, (ii), the phrase “ unramified over the cusps of ...” should be replaced by the unramified over the cusps as well as over the generic points of the phrase “ irreducible components of the special fibers of the stable models of ...”. Also, the phrase “tempered coverings of the underlying ...” should be replaced by the phrase “tempered admissible coverings of the underlying ...”. In a word, the thrust of both the original text and the slight modifications just ell ” is to be defined to be just large enough to discussed is that the monoid “Φ W include precisely those divisors which are necessary in order to treat the theta functions that appear in [EtTh]. ∈ Let v Example 3.3. Frobenioids at Good Nonarchimedean Primes. ⋂ non good V V . Then: (i) Write def def 0 0  D ( B ) = ; ) X K B = ( D v v v − → v

78 78 SHINICHI MOCHIZUKI  may be naturally regarded [by pulling back finite ́ etale coverings [cf. 0]. Thus, § D v )] as a → full subcategory K Spec( X via the structure morphism v → − v  D ⊆D v v  ,whichis D D →D , and we have a natural functor of left-adjoint to the natural v v v   inclusion functor D [cf. [FrdII], Example 1.3, (ii)]. For Spec( ↪ L ) ∈ Ob( D →D ) v v v def ×   L [i.e., K is a finite separable extension of O = O / O ) ], write ord( as in [FrdII], v L L L 0] Example 1.1, (i). Thus, the assignment [cf. §  pf → )  : Spec( ord( O ) L Φ C v L  on [ D monoid determines a , hence, by pull-back via the natural functor D Φ → C v v v  ,on] D ; the assignment D v v pf   Φ ) ) → ord( Z L : Spec(  )( ⊆ ord( O )  C p L v v ⊆ absolutely primitive [cf. [FrdII], Example 1.1, (ii)] submonoid Φ determines an  C v  | ; these monoids Φ p on D ,Φ determine -adic Frobenioids Φ   C v C C D v v v v v  ⊆C C v v [cf. [FrdII], Example 1.1, (ii), where we take “Λ” to be Z ], whose base categories   D [in a fashion compatible with the natural inclusion D , ⊆D ], D are given by v v v v respectively. Also, we shall write def = C F v v [cf. the notation of Example 3.2, (i)]. Finally, let us observe that the element   Z characteristic splitting determines a ∈ ⊆O p v p K v v  τ v def     ,τ F [cf. [FrdII], Theorem 1.2, (v)]. Write split C C ) for the resulting on =( v v v v Frobenioid . and considered additively of (i) )fortheelement p (ii) Next, let us write log( p v v  consider the monoid on D v ×  )) p ( − )= O ( log( · N ( − ) × O  v  C C v v  [cf. [FrdI], Proposition 2.2]. By replacing “log( p )” by the formal C associated to v v log(Θ ) )”, we obtain a monoid ) · log(Θ ) = log( p p symbol “log( v v def ×  ) p ( − ) log(Θ = O )) log( · · ( − ) × ( N O Θ v Θ C C v v

79 ̈ INTER-UNIVERSAL TEICHM 79 ULLER THEORY I def × ×  O − ) [i.e., where = O and which ( O to ( − )], which is naturally isomorphic Θ   C C C v v v  O arises as the monoid “ )” of [FrdI], Proposition 2.2, associated to some -adic − ( p v def  Θ Θ D with base category = D characteristic splitting equipped with a Frobenioid C v v v Θ τ determined by log( p ) · log(Θ ). In particular, we have a natural equivalence v v ∼  Θ F →F v v def Θ Θ Θ C ,τ F —where )—of split Frobenioids . =( v v v (iii) Here, it is useful to recall that  (a) the subcategory D ⊆D may be reconstructed category-theoretically v v [cf. [AbsAnab], Lemma 1.3.8]; from D v  Θ D (b) the category D )maybe reconstructed category-theoretically (respectively, v v Θ  C from (respectively, C ) [cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem v v 1.2, (i); [FrdII], Example 1.3, (i); [AbsAnab], Theorem 1.1.1, (ii)]; C may be reconstructed category-theoretically from F = (c) the category D v v v [cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem 1.2, (i); [FrdII], Example 1.3, (i); [AbsAnab], Lemma 1.3.1]. Note that it follows immediately from the category-theoreticity of the divisor monoid [cf. [FrdI], Corollary 4.11, (iii); [FrdII], Theorem 1.2, (i)], together with (a), Φ C v  C (c), and the definition of [cf. also [AbsAnab], Proposition 1.2.1, (v)], that v  reconstructed category-theoretically from may be . F C (d) v v algorithmically constructed field structure on the image Finally, by applying the Kummer map of the of [AbsTopIII], Proposition 3.2, (iii) [cf. Remark 3.1.2; Re-  O ”of p mark 3.3.2 below], it follows that one may construct the element “ v K v  F may be , hence that the characteristic splitting τ category-theoretically from v v . [Here, we recall that the curve X is F reconstructed category-theoretically from F v “of strictly Belyi type” — cf. [AbsTopIII], Remark 2.8.3.] In particular,  Θ F reconstruct the split Frobenioids (e) one may F , category-theoretically v v F from . v A similar remark to Remark 3.2.1 [i.e., concerning the phrase Remark 3.3.1.  ] applies to the Frobenioids C “reconstructed category-theoretically” , C constructed v v in Example 3.3.  Remark 3.3.2. Note that the p -adic Frobenioid C ) of Example (respectively, C v v v 3.3, (i), consists of essentially the same data as an “MLF-Galois TM -pairofstrictly Belyi type” (respectively, “MLF-Galois TM -pair of mono-analytic type” ), in the sense of [AbsTopIII], Definition 3.1, (ii) [cf. [AbsTopIII], Remark 3.1.1]. A similar

80 80 SHINICHI MOCHIZUKI  p -adic Frobenioid (respectively, C remark applies to the C ) of Example 3.2, (iii), v v v (iv) [cf. [AbsTopIII], Remark 3.1.3]. arc . Then: ∈ V Let v Example 3.4. Frobenioids at Archimedean Primes. (i) Write C C , X , , C , , X X v v v v → → − − v v [cf. [AbsTopIII], Definition 2.1, (i); [AbsTopIII], for the Aut -holomorphic orbispaces C , X , , X Remark 2.1.1] determined, respectively, by the hyperbolic orbicurves K K K , X complex , C C , ,wehavea at } .Thus,for  ∈{ X X , C , , X C , v C v v v v K − → − − → − → → v K K v [i.e., a “CAF” — cf. 0] § archimedean topological field A  [cf. [AbsTopIII], Definition 4.1, (i)] which may be algorithmically constructed from def 0 A [cf. Remark 3.4.3, (i), below]. Next, let us write = } \{ A ;write    def D = X v − → v and C v for the archimedean Frobenioid as in [FrdII], Example 3.3, (ii) [i.e., “ C ”of loc. cit. ], wherewetakethe base category D ”of loc. cit. ] to be the one-morphism [i.e., “ among the pseudo- linear morphisms ). Thus, the K category determined by Spec( v unique isomorphisms [cf. [FrdI], Definition 1.3, terminal objects of C determine  ( )”—wherewerecall O topological monoids − “ (iii), (c)] among the respective [cf. [FrdI], Theorem 3.4, (iii); [FrdII], Theorem 3.6, (i), (vii)] that these topological from C . In particular, it makes monoids may be reconstructed category-theoretically  ×  C C )”, “ O )”. Moreover, we observe that, by ( C ( ) ⊆O ( O sense to write “ v v v natural isomorphism construction, there is a ∼   ( →O C ) O v K v of topological monoids . Thus, one may also think of C “Frobenioid-theoretic as a v  [cf. [AbsTopIII], Remark 4.1.1]. representation” of the topological monoid O K v ∼ natural topological isomorphism K Observe that there is a ,whichmaybe A → D v v  restricted to O to obtain an inclusion of topological monoids K v  κ →A ( C ) ↪ O : v D v v — which we shall refer to as the Kummer structure on C [cf. Remark 3.4.2 below]. v Write def ) , =( C ,κ D F v v v v [cf. Example 3.2, (i); Example 3.3, (i)].

81 ̈ INTER-UNIVERSAL TEICHM 81 ULLER THEORY I  TM “split topological monoids” of [AbsTopIII], (ii) Next, recall the category of → − ( C ) consist of a topolog- C, Definition 5.6, (i) — i.e., the category whose objects − →  C and a topological submonoid C ⊆ [neces- O isomorphic to ical monoid C C × × C , ↪ → C [where C ] such that the natural inclusions R sarily isomorphic to 0 ≥ 1 , denotes the topological submonoid of invert- S which is necessarily isomorphic to − → → − ∼ × × → C of topological C ible elements C ], C↪ → C determine an isomorphism → − → − C ) are isomorphisms of topo- , → C C , ( ) C monoids, and whose morphisms ( 2 2 1 1 → − → − ∼ ∼ C → C C . Note that the that induce isomorphisms → C logical monoids 1 2 1 2  , A .Write determine, in a natural way, objects of TM K CAF’s v D v  τ v def  for the resulting characteristic splitting of the Frobenioid = C C , i.e., so that we v v     may think of the pair ( O ) ,τ C ; ) as the object of TM ( determined by K v v v  D v  for the object of determined by A TM ; D v def     F =( , D C ) ,τ v v v v       C for the [ordered] consisting of triple ( D ) . Thus, the object ( O ,and , C τ ,τ ) v v v v v      to D isomorphic .Moreover, C is algorith- (respectively, D )maybe F ; of TM v v v v F mically reconstructed from D ; F ). (respectively, v v v K ∈ [cf. § 0] may be thought of as a(n) [non- (iii) Next, let us observe that p v v [i.e., the factor denoted by a “ → ”in Φ noncompact factor identity] element of the  C v      ( C ] of the object ( TM O ,τ . This noncompact factor )of ) TM the definition of v v is isomorphic, as a topological monoid, to R ;letuswriteΦ additively Φ   ≥ 0 C C v v and denote by log( p ) the element of Φ p determined by . Thus, relative to  v v C v on Φ )isa p , it follows that log( R the natural action [by multiplication!] of  ≥ v 0 C v new topological monoid . In particular, we may form a of Φ generator  C v def Θ Φ R · ) p = ) · log(Θ log( 0 v ≥ C v ) log(Θ isomorphic to R formal symbol “log( p )”. ) · log(Θ ) = log( p that is generated by a v v 0 ≥ ×    Moreover, if we denote by O the compact factor of the object ( O )of ( C ) ,τ  v v C v def  × × Θ Θ , and set C O O ), isomor- = ,τ , then we obtain a new split Frobenioid ( TM Θ  v v C C v v   ), such that ,τ C phic to ( v v × Θ  Θ ( C )= O Φ × O Θ C v C v v

82 82 SHINICHI MOCHIZUKI natural isomorphism of split Frobe- — where we note that this equality gives rise to a ∼   Θ Θ ,τ → ( C ,τ ) )”. In ), obtained by “forgetting the formal symbol log(Θ ( nioids C v v v v particular, we thus obtain a natural isomorphism ∼  Θ F →F v v def Θ Θ Θ Θ Θ ,τ , D C , F —wherewewrite =( ) for the [ordered] triple consisting of C v v v v v def Θ Θ Θ  D algorithmically reconstructed = D . Finally, we observe that F , τ may be v v v v from F . v Remark 3.4.1. A similar remark to Remark 3.2.1 [i.e., concerning the phrase “algorithmically recon- “reconstructed category-theoretically” ] applies to the phrase that was applied in the discussion of Example 3.4. structed” Remark 3.4.2. Kummer structure One way to think of the  ↪ : ( C →A ) O κ v v D v discussed in Example 3.4, (i), is as follows. In the terminology of [AbsTopIII], Def- inition 2.1, (i), (iv), the structure of CAF on determines, via pull-back by , A κ v D v   gp ), together O C ( C of ) ( O an Aut on the groupification -holomorphic structure v v  gp co-holomorphicization with a [tautological!] O ( C →A ) . Conversely, if one D v v starts with this Aut-holomorphic structure on [the groupification of] the topological  gp  ,then ) ), together with the co-holomorphicization O →A ( C ( C monoid O v D v v one verifies immediately that one may recover the inclusion of topological monoids κ . [Indeed, this follows immediately from [AbsTopIII], Corollary 2.3, together v with the elementary fact that every holomorphic automorphism of the complex Lie × ≤ that preserves the submonoid of elements of norm 1 is equal to the C group .] That is to say, in summary, identity the Kummer structure κ is completely equivalent to the collection v ]on of data consisting of the Aut -holomorphic structure [induced by κ v   gp O the groupification ) O ( C of ), together with the co-holomorphi- C ( v v gp  cization [induced by κ O ( C ) ] →A . v D v v The significance of thinking of Kummer structures in this way lies in the observation that [unlike inclusions of topological monoids!] the induced by κ co-holomorphicization is log- with the compatible v arithm operation discussed in [AbsTopIII], Corollary 4.5. Indeed, this observation may be thought of as a rough summary of a substantial portion of the content of [AbsTopIII], Corollary 4.5. Put another way, thinking of Kummer structures in terms of co-holomorphicizations allows one to out separate the portion of the structures involved that is not compatible with this logarithm operation — i.e., the monoid structures! — from the portion of the structures involved that is compatible with this logarithm operation — i.e., the tautological co-holomorphicization .

83 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 83 Remark 3.4.3. × A (i) In the notation of Example 3.4, write for the topological group ⊆ A   × of [i.e., of elements of norm 1] of the CAF units [so A is noncanonically iso- A   μ × 1 torsion elements ]; ⊆A A for the subgroup of morphic to the unit circle S   μ Q -holomorphic space Z ]; E is noncanonically isomorphic to for the Aut / A [so v  [cf. [AbsTopIII], Definition 2.1, (i)] determined by the elliptic curve obtained by A construction v . Now recall from the ” in [AbsTopIII], of “ at X compactifying  K Corollary 2.7 [cf. also [AbsTopIII], Definition 4.1, (i)] via the technique of “holo- , that one has a natural isomorphism of CAF’s morphic elliptic cuspidalization” ∼ A A A → = X D X v v − → v A — which may be used to “identify” . Indeed, thinking of = A with A D X X v v → − v “ -approach” Θ = A A ” is natural from the point of view of the ”as“ A “ X X D v v → − v × discussed in Remark 3.1.2, (ii). Moreover, by allowing A to “act” [cf. the algo- X v rithm discussed in [AbsTopIII], Corollary 2.7, (e)] on points in a sufficiently small of [but not equal to !] a given point “ x ”of E neighborhood , one may regard the v × “circle” A as a x in a suitable small deformation retract of the complement of X v E x neighborhood of in .Inparticular, v Θ -approach” discussed in Remark 3.1.2, from the point of view of the “ (ii), it is natural to think of “ ” and to regard A = A A ”as“ D X X v v → − v μ × Q / Z Hom( A , ) = Hom( Q / ) Z A , X X v v ̂ Z [a profinite group which is noncanonically isomorphic to ] as the result of identifying the cuspidal inertia groups of the various points “ x ”of E v — cf. discussion of the cuspidal inertia groups “ I ” in [AbsTopIII], Proposition x 1.4, (i), (ii). Indeed, this interpretation of via cuspidal inertia groups A A = D X v − → v archimedean version of the “ may be thought of as a sort of -approach” discussed Θ in Remark 3.1.2, (ii). (ii) We observe that just as the theory of elliptic cuspidalization [cf. [AbsTopII], Example 3.2; [AbsTopII], Corollaries 3.3, 3.4] admits a straightforward holomorphic analogue , i.e., the theory of “holomorphic elliptic cuspidalization” [cf. [AbsTopIII], Corollary 2.7] referred to in (i) above, the theory of Belyi cuspidalization [cf. [Ab- sTopII], Example 3.6; [AbsTopII], Corollaries 3.7, 3.8; [AbsTopIII], Remark 2.8.3] admits a straightforward holomorphic analogue , i.e., a theory of “holomorphic Belyi cuspidalization” . We leave the routine details to the reader. Here, we ob- serve that one immediate consequence of such “holomorphic Belyi cuspidalizations” may be stated as follows: the set of NF-points [i.e., points defined over a number field ]ofthe may be underlying topological space of the Aut-holomorphic space D v reconstructed via a functorial algorithm from the [abstract] Aut- . holomorphic space D v

84 84 SHINICHI MOCHIZUKI Example 3.5. Global Realified Frobenioids. (i) Write  C mod realification for the [cf. [FrdI], Theorem 6.4, (ii)] of the Frobenioid of [FrdI], Example F number field 6.3 [cf. also Remark 3.1.5 of the present paper], associated to the mod [so and the trivial Galois extension [i.e., the Galois extension of degree 1] of F mod  is, in the terminology of [FrdI], equivalent to a one- C the base category of mod   divisor monoid Φ morphism category ]. Thus, the C of may be thought of C mod mod  ) as a single abstract monoid, whose C , which we denote Prime( set of primes mod [cf. the discussion natural bijective correspondence with V [cf. [FrdI], § 0], is in mod   of Φ corresponding of [FrdI], Example 6.3]. Moreover, the submonoid Φ C C ,v mod mod  pf ∼ v V to ∈ ) [i.e., to is naturally isomorphic R to ord( O ) ⊗ R ( = 0 ≥ ≥ 0 mod ( F ) v mod  arc ∼ determines an element V ∈ p )( ]. In particular, v R )if O ord( = 0 ≥ v mod ) F ( v mod   log .Write ) ∈ Φ ( p . v that corresponds to v ∈ V for the element of V v ,v C mod mod ⋂ non arc bad good V V belongs to ,or V , , V v Then observe that regardless of whether  rlf of the divisor monoid Φ C [which, as is easily verified, of Φ realification the   C v C v v constant monoid over the corresponding base category] may be regarded as a is a rlf ∈ for the element Φ .Writelog ) ( p isomorphic to single abstract monoid R  v 0 ≥ Φ C v defined by p and v rlf   : C ) C → ( C ρ mod v v natural restriction functor [cf. the theory of poly-Frobenioids developed in for the  [cf. [FrdI], Proposition 5.3]. 5] to the realification [FrdII], § of the Frobenioid C v C Thus, one verifies immediately that is determined, up to isomorphism, by the ρ v ] [which are isomorphic to R isomorphism of topological monoids 0 ≥ ∼ rlf  ρ Φ :Φ →  v C ,v C v mod induced by C — which, by considering the natural “volume interpretations” of ρ v the arithmetic divisors involved, is easily computed to be given by the assignment  1 ( p p )  → ). ( log log v v Φ mod F ] [ ) :( K v v mod (ii) In a similar vein, one may construct a Θ -version” [i.e., as in Examples “ 3.2, (v); 3.3, (ii); 3.4, (iii)] of the various data constructed in (i). That is to say, we set def   Φ · ) =Φ log(Θ C C tht mod  — i.e., an isomorphic copy of Φ log(Θ ). This generated by a formal symbol C mod   , equipped with a thus determines a Frobenioid C natural equivalence monoid Φ C tht tht ∼ ∼    ∈ →C and a natural bijection Prime( C v .For ) V → of categories C mod mod tht tht    V , the element log thus determines ) of the submonoid Φ ( p ⊆ Φ mod v C C ,v mod mod mod    ( p ∈ ) · log(Θ ) of a submonoid Φ v V .Write for the ⊆ Φ an element log v C C ,v mod tht tht rlf . Then the realification Φ v that corresponds to of the divisor monoid V element of Θ C v Θ Θ Φ of C [which, as is easily verified, is a constant monoid over the corresponding C v v

85 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 85 R . base category] may be regarded as a isomorphic to single abstract monoid ≥ 0 Write  rlf Θ Θ C → ( C C : ) ρ tht v v [cf. (i) above; the theory of poly-Frobenioids de- for the natural restriction functor Θ [cf. [FrdI], Proposi- velopedin[FrdII], § 5] to the realification of the Frobenioid C v Θ is determined, up to isomorphism, C tion 5.3]. Thus, one verifies immediately that ρ v R [which are isomorphic to isomorphism of topological monoids by the ] ≥ 0 ∼ rlf Θ  :Φ → Φ ρ Θ ,v C v C v tht good rlf Θ C .If v ∈ V for the element ,thenwritelog induced by ( p Φ ) · log(Θ ) ∈ Θ v ρ Φ C v v  Θ determined by log ( ρ p ( is given by the assignment log ); thus, [cf. (i)] · ) p v v Φ v mod bad 1 → V , then let us )  ∈ v ). On the other hand, if log(Θ · ) log p ( log(Θ v Φ ) :( K [ F ] v v mod write rlf log ) ∈ Φ (Θ Θ Φ C v v p [cf. Example 3.2, (v)] and log ) for the ( for the element determined by Θ v Φ v constant section of Φ )” of Example determined by p q [cf. the notation “log ( v C Φ v v 3.2, (iv)]; in particular, it makes sense to write log ( p Q / log ( q ;thus,[cf. ) ∈ ) v > 0 Φ Φ v Θ is given by the assignment ρ (i)] v log ) (Θ Φ log ) ( p v v Φ  log · )  ( p → · log(Θ ) v mod q ] K ) :( F [ ) log ( mod v v Φ v v — cf. Remark 3.5.1, (i), below. Note that, for arbitrary V ρ , , the various ∈ v ∼ ∼ Θ Θ    ρ 0]. with the →C compatible § , C are natural isomorphisms C →C [cf. v v v mod tht natural isomorphism between collections of data This fact may be expressed as a V [consisting of a category, a bijection of sets, a collection of data indexed by ,and ] V a collection of isomorphisms indexed by ∼   → F F mod tht —wherewewrite ∼ def     F , Prime( C =( } ) C → V , {F , ρ } { ) V V v ∈ v ∈ v v mod mod mod ∼ def Θ  Θ   { , C ) , {F } ρ } =( F Prime( ) C → V , v V ∈ V ∈ v v v tht tht tht ∼ V ]; cf. Remark 3.5.2 below. → V [and we apply the natural bijection mod (iii) One may also construct a “ D -version” — which, from the point of view of the theory of [AbsTopIII], one may also think of as a “log-shell version” —ofthe various data constructed in (i), (ii). To this end, we write  D mod

86 86 SHINICHI MOCHIZUKI   . Thus, one may associate to D C for a [i.e., another] copy of various objects mod mod ∼ D     Φ → V ] ,log Φ V ( p ∈ ) ∈ ,Prime( v ) [for D ⊆ Φ mod mod v D D D ,v mod mod mod mod mod  tautological under the C that map to the corresponding objects associated to mod ∼   equivalence of categories C .Write v V for the element of V that →D ∈ mod mod non V ; then let us recall from [AbsTopIII], ∈ v v corresponds to . Next, suppose that  is the absolute D Proposition 5.8, (iii), that [since the profinite group associated to v  D Galois group of an MLF] one may construct algorithmically from topological a v monoid isomorphic to R ≥ 0  ( R ) v 0 ≥ [i.e., the topological monoid determined by the nonnegative elements of the ordered G )” of loc. cit. ] equipped with a distinguished “Frobenius ( R topological group “ non  e ,thenwe K ) of the MLF ;if absolute ramification index is the R element” ∈ ( v v v ≥ 0 D  ( p ) ∈ ( R for the result of multiplying this Frobenius element ) shall write log v v Φ ≥ 0 arc e by [the positive real number] v ∈ V . Next, suppose that ; then let us recall v   )] ∈ Ob( TM from [AbsTopIII], Proposition 5.8, (vi), that [since, by definition, D v  from D one may construct algorithmically a R isomorphic to topological monoid ≥ 0 v  ( R ) v 0 ≥ [i.e., the topological monoid determined by the nonnegative elements of the ordered distinguished “Frobenius ] equipped with a loc. cit. ( G )” of R topological group “ arc D   ( ; we shall write log R ) for the result of dividing this p ) ) ∈ ( ∈ element” R ( v v v Φ ≥ 0 ≥ 0 v . In particular, for every π Frobenius element by [the positive real number] 2 V , ∈ [which are we obtain a uniquely determined isomorphism of topological monoids ] R isomorphic to ≥ 0 ∼  D  :Φ → ( R ) ρ v D ,v 0 ≥ v mod D D 1 p data → p ( [consisting of  ) ). Thus, we obtain log by assigning log ( v v Φ mod [ ) :( ] K F v v mod a Frobenioid, a bijection of sets, a collection of data indexed by V , and a collection ] V of isomorphisms indexed by def ∼     D F V =( ) ) → D , {D , } Prime( , { ρ D } ∈ V v ∈ v V v mod mod v D ∼ [where we apply the natural bijection V V → ], which, by [AbsTopIII], Proposi- mod  . } data {D tion 5.8, (iii), (vi), may be reconstructed algorithmically from the v V ∈ v Remark 3.5.1. (i) The formal symbol “log(Θ )” may be thought of as the result of identifying varies over the elements of v (Θ )”, as ) / log q ( the various formal quotients “log Φ Φ v v bad V .   C (ii) The global Frobenioids , C of Example 3.5 may be thought of as tht mod “devices for currency exchange” between the various “local currencies” constituted ∈ V . by the divisor monoids at the various v

87 ̈ INTER-UNIVERSAL TEICHM 87 ULLER THEORY I   F F (iii) One may also formulate the data contained in , via the language tht mod § 5, but we shall not pursue this topic in poly-Frobenioids of as developed in [FrdII], the present series of papers. In Example 3.5, as well as in the following discussion, we shall Remark 3.5.2. often speak of , relative to the following con- “isomorphisms of collections of data” ventions. evident compati- (i) Such isomorphisms are always assumed to satisfy various , relative to the various relationships stipulated between the various bility conditions constituent data, whose explicit mention we shall omit for the sake of simplicity. cat- (ii) In situations where the collections of data consist partially of various egories , the portion of the “isomorphism of collections of data” involving corre- sponding categories is to be understood as an isomorphism class of equivalences of § categories [cf. 0]. bad ) , V F/F, X , ,l,C V , Fix a collection of initial ( Definition 3.6. -data Θ F K mod as in Definition 3.1. In the following, we shall use the various notations introduced in Definition 3.1 for various objects associated to this initial Θ-data. Then we define aΘ -Hodge theater [relative to the given initial Θ-data] to be a collection of data Θ † †  † F } { , =( F HT ) V v ∈ mod v that satisfies the following conditions: non † (a) If v F V ,then category which admits an equivalence of cate- ∈ is a v ∼ † F →F [where F is as in Examples 3.2, (i); 3.3, (i)]. In partic- gories v v v † ular, F admits a natural Frobenioid structure [cf. [FrdI], Corollary 4.11, v (iv)], which may be constructed solely from the category-theoretic struc- Θ Θ † †  † † †  † D for the objects , D constructed F , , F D , .Write F ture of v v v v v v † category-theoretically from F that correspond to the objects without a v † ” discussed in Examples 3.2, 3.3 [cf., especially, Examples 3.2, (vi); 3.3, “ (iii)]. arc † † † † † v (b) If collection of data ( , C ,then , ∈ D C F V κ is a )—where v v v v v † is an C of Example 3.4, (i); equivalent to the category D is a category v v †  † is an inclusion ↪ : O →A ( κ C ) Aut ;and -holomorphic orbispace † v v D v of topological monoids, which we shall refer to as the Kummer structure † C — such that there exists an isomorphism of collections of data on v ∼ † †   † Θ † D F [where , F D , F , →F is as in Example 3.4, (i)]. Write v v v v v v † Θ † for the objects constructed algorithmically from F F that correspond v v † ” discussed in Example 3.4, (ii), (iii). to the objects without a “ †  (c) is a collection of data F mod ∼ †   † †  † } C } ) , → V , { ρ F C { Prime( , ) ( v V ∈ ∈ V v v mod mod v

88 88 SHINICHI MOCHIZUKI  † is a category which admits an equivalence of categories —where C mod ∼  †  †  C [which implies that →C admits a natural category-theore- C mod mod mod tically constructible — cf. [FrdI], Corollary 4.11, Frobenioid structure ∼  † (iv); [FrdI], Theorem 6.4, (i)]; Prime( → V is a bijection of sets, ) C mod  † where we write Prime( C ) for the set of primes constructed from the mod † †   C F is as discussed in [cf. [FrdI], Theorem 6.4, (iii)]; category v mod ∼ † rlf  ρ Φ [where we use notation as in the → :Φ (a), (b) above; † †  v C ,v C mod v discussion of Example 3.5, (i)] is an isomorphism of topological monoids . Moreover, we require that there exist an isomorphism of collections of data ∼ †   †   †  F F is as in Example 3.5, (ii)]. Write [where F F F , → D tht mod mod mod †  for the objects constructed algorithmically from that correspond to F mod † the objects without a “ ” discussed in Example 3.5, (ii), (iii). Remark 3.6.1. When we discuss various collections of Θ-Hodge theaters, labeled ”inplaceofa“ †  by some symbol “ ”, we shall apply the notation of Definition 3.6 with “ † ” replaced by “  ” to denote the various objects associated to the Θ-Hodge  ”. theater labeled by “ Θ Θ ‡ † -Hodge theaters and HT are Θ HT , then there is an Remark 3.6.2. If ∼ Θ Θ † ‡ HT [cf. Remark → HT -Hodge theaters isomorphism of Θ evident notion of 3.5.2]. We leave the routine details to the interested reader. Θ -Links Between Θ -Hodge Theaters) Fix a collection of Corollary 3.7. ( bad Θ -data ( initial ,l,C , V , V F/F, X as in Definition 3.1. Let , ) F K mod Θ Θ † †  ‡  ‡ ‡ † { , HT F F , ); ) HT } =( { F F =( } v V V ∈ v ∈ mod mod v v Θ [relative to the given initial Θ -data]. Then: be -Hodge theaters ( Θ -Link) The full poly-isomorphism [cf. § 0] between collections of data (i) [cf. Remark 3.5.2] ∼   † ‡ → F F tht mod is nonempty [cf. Remark 3.7.1 below]. We shall refer to this full poly-isomorphism as the Θ -link Θ Θ Θ † ‡ HT −→ HT Θ Θ † ‡ to HT . from HT  V . Recall the ∈ tautological isomor- v ”) Let D (ii) (Preservation of “ ∼    Θ phisms → for D — i.e., which arise from the definitions when ‡ D  = † , v v good V [cf. Examples 3.3, (ii); 3.4, (iii)], and which arise from a natural product ∈ v bad V ∈ . Then we obtain a composite [full] v functor [cf. Example 3.2, (v)] when poly-isomorphism ∼ ∼  ‡  † Θ † D → → D D v v v by composing the tautological isomorphism just mentioned with the poly-isomorphism induced by the Θ -link poly-isomorphism of (i).

89 ̈ INTER-UNIVERSAL TEICHM 89 ULLER THEORY I × (Preservation of “ ”) Let v ∈ V . Recall the tautological isomor- (iii) O ∼ × × phisms O ”] for →O — i.e., † =  , [where we omit the notation “ ( − ) ‡    Θ C C v v good v which arise from the definitions when ∈ [cf. Examples 3.3, (ii); 3.4, (iii)], V and which are induced by the natural product functor [cf. Example 3.2, (v)] when bad V ∈ . Then, relative to the corresponding composite isomorphism of (ii), we v obtain a composite [full] poly-isomorphism ∼ ∼ × × × →O →O O  Θ † ‡  † C C C v v v by composing the tautological isomorphism just mentioned with the poly-isomorphism induced by the of (i). -link poly-isomorphism Θ Proof. The various assertions of Corollary 3.7 follow immediately from the defini- tions and the discussion of Examples 3.2, 3.3, 3.4, and 3.5. © many distinct isomor- One verifies immediately that there exist Remark 3.7.1. ∼ ‡  †  F F as in Corollary 3.7, (i), none of which is conferred a “dis- → phisms mod tht ∼   status, i.e., in the fashion of the F “natural isomorphism tinguished” → ” F mod tht discussed in Example 3.5, (ii). The following result follows formally from Corollary 3.7. Corollary 3.8. (Frobenius-pictures of Θ -Hodge Theaters) Fix a collection Θ n of initial -data as in Corollary 3.7. Let { Θ be a collection of distinct } HT n Z ∈ indexed by the integers. Then by applying Corollary 3.7, (i), Θ -Hodge theaters def def Θ Θ Θ Θ n n ( ‡ +1) † , we obtain an HT , HT = HT HT = infinite chain with Θ Θ Θ Θ Θ Θ Θ 1) n n ( ( n +1) − HT −→ HT HT −→ −→ −→ ... ... of -linked Θ -Hodge theaters . This infinite chain may be represented symboli- Θ  oriented graph cally as an [cf. [AbsTopIII], § 0] Γ →•→•→•→ ... ... Θ — i.e., where the arrows correspond to the “ −→ ’s”, and the “ • ’s” correspond to the Θ n  “ translation ”. This oriented graph admits a natural action by HT — i.e., a Γ Z symmetry — but it does not admit arbitrary permutation symmetries .For  instance, does not admit an automorphism that switches two adjacent vertices, Γ but leaves the remaining vertices fixed. Put another way, from the point of view of the discussion of [FrdI], § I4, the mathematical structure constituted by this infinite chain is “Frobenius-like” ,or “order-conscious” . It is for this reason that we shall refer to this infinite chain in the following discussion as the Frobenius- picture .

90 90 SHINICHI MOCHIZUKI Remark 3.8.1. (i) Perhaps the central defining aspect of the Frobenius-picture is the fact that the Θ-link maps n ( n +1) → q Θ  v v bad v ∈ V — cf. the discussion of Example 3.2, (v)]. From this point of [i.e., where view, the Frobenius-picture may be depicted as in Fig. 3.1 below — i.e., each box is a Θ-Hodge theater; the “ ” may be thought of as denoting the scheme theory  ”; the “- - - -” denotes the Θ-link. ”and“Θ q that lies between “ v v n +1) n n ( ( n +1)  q Θ Θ q  ---- ---- ---- v v v v ... ... +1) ( n n  → q Θ v v Fig. 3.1: Frobenius-picture of Θ-Hodge theaters (ii) It is perhaps not surprising [cf. the theory of [FrdI]] that the Frobenius- portion [i.e., “ q divisor monoid picture involves, in an essential way, the ”and v “Θ ”] of the various Frobenioids that appear in a Θ-Hodge theater. Put another v way, it is as if the of the divisor monoid portion of the “Frobenius-like nature” Frobenioids involved induces the “Frobenius-like nature” of the Frobenius- picture. , the isomorphisms V ∈ v observe By contrast, that for ∼ ∼ ∼  n ( n +1)  ... ... → D D → → v v −  ) ( as being only D of Corollary 3.7, (ii), imply that if one thinks of the various v ,then up to isomorphism known − ) (  as a sort of constant invariant of the various D one may regard v -Hodge theaters that constitute the Frobenius-picture Θ observation — cf. Remark 3.9.1 below. This is the starting point of the theory of the ́ etale-picture [cf. Corollary 3.9, (i), below]. Note that by Corollary 3.7, (iii), we also obtain isomorphisms ∼ ∼ ∼ × × ... → →O →O ... n  n +1) (  C C v v ( − )  lying over the isomorphisms involving the “ D ” discussed above. v non ) − (  v (iii) In the situation of (ii), suppose that is simply D .Then V ∈ v the category of connected objects of the Galois category associated to the profinite ( − )  G “ as representing D up to . That is to say, one may think of G group v v v

91 ̈ INTER-UNIVERSAL TEICHM 91 ULLER THEORY I n . Then each isomorphism” represents an “isomorph of the topological group D v , by n , which is regarded as an extension of some isomorph of G labeled that Π v v of the n ” . In particular, the quotients corresponding to G is independent of v Θ n copies of Π HT for different n are only related to one another that arise from v isomorphism. Thus, from the point of view of the theory of via some indeterminate gives [AbsTopIII] [cf. [AbsTopIII], § I3; [AbsTopIII], Remark 5.10.2, (ii)], each Π v well-defined ring structure rise to a “holomorphic structure” —whichis — i.e., a obliterated by the indeterminate isomorphism between the quotient isomorphs of Θ n arising from HT for distinct n . G v arc ( − )  D ∈ V is an object .Then v (iv) In the situation of (ii), suppose that v  n TM of D , ; each “isomorph of the represents an -holomorphic orbispace X Aut v − → v gives A n , whose associated [complex archimedean] topological field by labeled X − → v  D rise to an isomorph of independent of n ” . In particular, the various that is v Θ  n D isomorphs of for different associated to the copies of HT that arise from X v → − v indeterminate isomorphism. Thus, from are only related to one another via some n § I3; [AbsTopIII], the point of view of the theory of [AbsTopIII] [cf. [AbsTopIII], Remark 5.10.2, (ii)], each X — i.e., a well-defined ring structure gives rise to a → − v “holomorphic structure” —whichis obliterated by the indeterminate isomorphism Θ  n D between the isomorphs of arising from HT for distinct n . v The discussion of Remark 3.8.1, (iii), (iv), may be summarized as follows. ́ Θ -Hodge Theaters) In the situation of Corollary 3.9. ( Etale-pictures of v Corollary 3.8, let V .Then: ∈ n D v | ... ... |  —— —— 1) +1) n − n ( ( D D D v v v | ... ... | ( n +2) D v ́ Fig. 3.2: Etale-picture of Θ-Hodge theaters

92 92 SHINICHI MOCHIZUKI n D v ... ... | | ×  —— ——  O D +1) n ( n − ( 1)  v C D D v v v | ... ... | ( n +2) D v ́ Fig. 3.3: Etale-picture plus units (i) We have a as in Fig. 3.2 above, which we refer to as the ́ etale- diagram picture . Here, each horizontal and vertical “— —” denotes the relationship be- non ) − (  tween D and , — i.e., an extension of topological groups when v ∈ V D v v  TM or the underlying object of arising from the associated topological field when arc ∈ — discussed in Remark 3.8.1, (iii), (iv). The ́ etale-picture [unlike the V v among the la- Frobenius-picture!] admits arbitrary permutation symmetries ∈ Z corresponding to the various Θ -Hodge theaters. Put another way, the n bels etale-picture may be thought of as a sort of ́ of the Frobenius- canonical splitting picture. diagram as in Fig. 3.3 above, obtained (ii) In a similar vein, we have a ×   ” in the middle of Fig. 3.2 by “ ”. Here, each  O D D by replacing the “  v v C v  ( − ) horizontal and vertical “— —” denotes the relationship between D D and v v non ×  D discussed in (i); when V ∈ O v ” denotes an isomorph , the notation “   v C v ×  D of the pair consisting of the category together with the group-like monoid O  v C v arc ×   v ∈ V on , the notation “ D D ” denotes an isomorph of the  O ;when  v v C v  ×  [which Ob( TM ) and the topological group O ∈ pair consisting of the object D  v C v  ]. Just as in D is isomorphic — but not canonically! — to the compact factor of v the case of (i), this diagram admits arbitrary permutation symmetries among n ∈ Z corresponding to the various Θ -Hodge theaters. the labels If one formulates things relative to the language of [AbsTopIII], Remark 3.9.1. ( − )  Definition 3.5, then D constitutes a core . Relative to the theory of [AbsTopIII], v § 5, this core is essentially the mono-analytic core discussed in [AbsTopIII], § I3; [AbsTopIII], Remark 5.10.2, (ii). Indeed, the symbol “  ” is intended — both in [AbsTopIII] and in the present series of papers! — as an abbreviation for the term “mono-analytic” .

93 ̈ INTER-UNIVERSAL TEICHM 93 ULLER THEORY I Whereas the Remark 3.9.2. of Corollary 3.9, (i), will remain valid ́ etale-picture throughout the development of the remainder of the theory of the present series of × ” that appear in Corollary 3.9, (ii), will ultimately cease O papers, the local units “  C v of various enhanced versions of the Frobenius-picture that to be a constant invariant will arise in the theory of [IUTchIII]. In a word, these enhancements revolve around the incorporation into each Hodge theater of the ’] “rotation of addition [i.e., ‘   ’]” in the style of the theory of [AbsTopIII]. and multiplication [i.e., ‘ Remark 3.9.3. (i) As discussed in [AbsTopIII], § I3; [AbsTopIII], Remark 5.10.2, (ii), the  fixed underly- may be thought of as a sort of } {D “mono-analytic core” v ∈ V v ing real-analytic surface associated to a number field on which various holo- are imposed. Then the Frobenius-picture in its entirety may morphic structures be thought of as a sort of Te- of the notion of a global arithmetic analogue in classical complex Teichm ̈ uller theory or, alternatively, as a ichm ̈ uller geodesic canonical liftings of global arithmetic analogue of the -adic Teichm ̈ uller theory p [cf. the discussion of [AbsTopIII], I5]. § uller theory, one of the (ii) Recall that in classical complex Teichm ̈ real di- two mensions of the surface is dilated as one moves along a Teichm ̈ uller geodesic, while the other of the two real dimensions is held fixed . In the case of the Frobenius- × ” correspond to the dimension that picture of Corollary 3.8, the local units “ O -dilations” held fixed are subject to “ Θ local value groups as one is , while the moves along the diagram constituted by the Frobenius-picture. Note that in order to construct such a mathematical structure in which the local units and local value independently , it is of crucial importance to avail oneself of groups are treated characteristic splittings that appear in the split Frobenioids of Ex- the various amples 3.2, 3.3, 3.4. Here, we note in passing that, in the case of Example 3.2, this splitting corresponds to the of the ́ etale theta “constant multiple rigidity” , which forms a central theme of the theory of [EtTh]. function (iii) In classical complex Teichm ̈ uller theory, the two real dimensions of the surface that are treated independently of one another correspond to the real and imaginary parts of the coordinate obtained by locally integrating the square root of a given square differential. In particular, it is of crucial importance in classical complex Teichm ̈ uller theory that these real and imaginary parts notbe“subjectto . In the case of the square root of a square differential, confusion with one another” multiplication the only indeterminacy that arises is indeterminacy with respect to − 1, an operation that satisfies the crucial property of preserving the real and by . By contrast, it is interesting to note that imaginary parts of a complex number if, for n ≥ 3, one attempts to construct Teichm ̈ uller deformations in the fashion of classical complex Teichm ̈ uller theory by means of coordinates obtained by n -th root of a given section of the n - locally integrating the th tensor power of the sheaf of differentials , then one must contend with an indeterminacy with respect to multiplication by an n -th root of unity , an operation that results in an essential confusion between the real and imaginary parts of a complex number .

94 94 SHINICHI MOCHIZUKI  Γ of Corollary 3.8 cor- (iv) Whereas linear movement along the oriented graph along a Teichm ̈ uller geodesic, the linear flow responds to the “rotation of addition ’] and multiplication [i.e., ‘ ’]” in the style of the theory of [AbsTopIII]  [i.e., ‘  — which will be incorporated into the theory of the present series of papers in in [IUTchIII] [cf. Remark 3.9.2] — corresponds to rotations around a fixed point the complex geometry arising from Teichm ̈ uller theory [cf., e.g., the discussion of [AbsTopIII], § I3; the hyperbolic geometry of the upper half-plane, regarded as the “Teichm ̈ uller space” of compact Riemann surfaces of genus 1]. Alternatively, in the p -adic Teichm ̈ uller theory, this “rotation of  and analogy with ” corresponds  to the — cf. the discussion of [Ab- Frobenius morphism in positive characteristic § I5. sTopIII], +1) n n (  q → Θ ” [cf. Remark Remark 3.9.4. At first glance, the assignment “ v v “conventional eval- 3.8.1, (i)] may strike the reader as being nothing more than a uation map” [i.e., of the theta function at a torsion point — cf. the discussion of Example 3.2, (iv)]. Although we shall ultimately be interested, in the theory of the present series of papers, in such “Hodge-Arakelov-style evaluation maps” [within a fixed Hodge theater!] of the theta function at torsion points” [cf. the theory of [IUTchII]], the Θ-link considered here from such con- differs quite fundamentally ventional evaluation maps in the following respect: n +1) ( belongs to a distinct scheme theory — i.e., the the value q v Θ +1) ( n -Hodge theater Θ scheme theory represented by the distinct — HT n q [which belongs to the scheme theory represented by the base from the v Θ n n -Hodge theater Θ ] over which the theta function Θ is constructed. HT v The distinctness of the ring/scheme theories of distinct Θ-Hodge theaters may be seen, for instance, in the indeterminacy of the isomorphism between the associated  obliterating , an indeterminacy which has the effect of the ring isomorphs of D v n D for “arithmetic holomorphic structure” — associated to structure — i.e., the v distinct n [cf. the discussion of Remark 3.8.1, (iii), (iv)].

95 ̈ INTER-UNIVERSAL TEICHM 95 ULLER THEORY I uller Theory Section 4: Multiplicative Combinatorial Teichm ̈ 4, we begin to prepare for the construction of the various In the present § 3 that will be made in § § “enhancements” to the Θ-Hodge theaters of 5. More of the — i.e., D ” precisely, in the present § 4, we discuss the “ combinatorial aspects of the portion “base category” in the terminology of the theory of Frobenioids, the — notions to be introduced in § 5 below. In a word, these combinatorial aspects revolve imposed upon the various number fields and “functorial dynamics” around the local fields involved by the “labels” def  × = } / {± 1 F F ∈ l l def   cardinality l =( is of -torsion l − 1) / 2 — of the l F — where we note that the set l “Hodge-Arakelov-theoretic points at which we intend to conduct, in [IUTchII], the evaluation” of the ́ etale theta function studied in [EtTh] [cf. Remarks 4.3.1; 4.3.2; 4.5.1, (v); 4.9.1, (i)]. In the following, we fix a collection of -data initial Θ bad , , V , V ,l,C F/F, X ) ( F mod K as in Definition 3.1; also, we shall use the various notations introduced in Definition 3.1 for various objects associated to this initial Θ-data. Definition 4.1. (i) We define a holomorphic base-prime-strip ,or -prime-strip , [relative to the D given initial Θ-data] to be a collection of data † † { = D D } v V v ∈ non † V category ,then that satisfies the following conditions: (a) if D v is a ∈ which v ∼ † D is as in Examples 3.2, (i); →D [where D admits an equivalence of categories v v v arc † ∈ ,then D V is an Aut -holomorphic orbispace such that there 3.3, (i)]; (b) if v v ∼ † exists an isomorphism of Aut-holomorphic orbispaces →D D [where D is as in v v v non † V D ,then π ) determines, in a functorial ( ∈ v Example 3.4, (i)]. Observe that if 1 v good ” group corresponding to “ ] C ∈ V v [in fact, profinite topological if fashion, a v good bad V ∈ ; [EtTh], Proposition 2.4, if v ], which contains V ∈ v [cf. Corollary 1.2 if 0 † † of this topological D ) ) as an open subgroup; thus, if we write for D − ( B ( π v 1 v † † → D D [cf. § 0]. In a similar vein, if natural morphism group, then we obtain a v v arc v admits a ,thensince X -core, a routine translation into the “language K V ∈ v → − v of Aut-holomorphic orbispaces” of the argument given in the proof of Corollary 1.2 † [cf. also [AbsTopIII], Corollary 2.4] reveals that D determines, in a functorial v † D corresponding to “ C ”, together with fashion, an Aut-holomorphic orbispace v v † † natural morphism a D → D of Aut -holomorphic orbispaces . Thus, in summary, v v one obtains a collection of data † † = { D D } v ∈ V v

96 96 SHINICHI MOCHIZUKI † . completely determined by D (ii) Suppose that we are in the situation of (i). Then observe that by applying the of [AbsTopI], Lemma 4.5 [cf., especially, [AbsTopI], group-theoretic algorithm Lemma 4.5, (v), as well as Remark 1.2.2, (ii), of the present paper], to construct the set of conjugacy classes of cuspidal decomposition groups of the topological non † D ) of a cofinal collection of )when ( ∈ V v , or by considering π − ( π group 0 v 1 “neighborhoods of infinity” [i.e., complements of compact subsets] of the underlying arc † when v ∈ D set of cusps , it makes sense to speak of the V topological space of v † † , then we define D D ; a similar observation applies to ,for v ∈ V .If v V ∈ of v v † † to be the set of cusps of D D that lie over a single a label class of cusps of v v [i.e., a cusp that arises from a nonzero element of the quotient “ Q ” “nonzero cusp” ”given ,l that appears in the definition of a “hyperbolic orbicurve of type (1 -tors) ± † ;write D in [EtTh], Definition 2.1] of v † LabCusp( D ) v † † D ) . Thus, for each D ∈ V , LabCusp( v of set of label classes of cusps for the v v  F admits a natural -torsor structure [i.e., which arises from the natural action of l × F on the quotient “ v ∈ V !] one Q ” of [EtTh], Definition 2.1]. Moreover, [for any l † D ,a canonical element may construct, solely from v † † η LabCusp( ∈ D ) v v  determined by “ ” [cf. the notation of Definition 3.1, (f)]. [Indeed, this follows v ⋂ non good bad V , from Corollary 1.2 for v ∈ V ∈ V , v from [EtTh], Corollary 2.9, for and from the evident translation into the “language of Aut-holomorphic orbispaces” arc .] ∈ V of Corollary 1.2 for v  -prime-strip ,[relative ,or mono-analytic base-prime-strip (iii) We define a D to the given initial Θ-data] to be a collection of data †   † D D = } { ∈ v V v non †  that satisfies the following conditions: (a) if ,then v D ∈ V is a category which v ∼    † D is as in Examples 3.2, [where D →D admits an equivalence of categories v v v arc   †  ,then D D ∈ V is an object of the category TM [so, if v (i); 3.3, (i)]; (b) if v v ∼  †   D is as in Example 3.4, (ii), then there exists an isomorphism in TM →D ]. v v  (iv) A - (respectively, D morphism of D -) prime-strips is defined to be a col- , between the various constituent objects of the lection of morphisms, indexed by V prime-strips. Following the conventions of § 0, one thus has a notion of capsules of   morphisms of capsules of D - (respectively, D and -) prime- -) D D - (respectively, † D , one may associate, in a natural way, strips . Note that to any D -prime-strip  †  -prime-strip D — which we shall refer to as the mono-analyticization of a D † D — by considering appropriate subcategories at the nonarchimedean primes [cf. Examples 3.2, (i), (vi); 3.3, (i), (iii)], or by applying the construction of Example 3.4, (ii), at the archimedean primes .

97 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 97 (v) Write def  0 D = B ( C ) K 0]. Then recall from [AbsTopIII], Theorem 1.9 [cf. Remark 3.1.2], that § [cf.  § ( D 0], )[cf. π there exists a group-theoretic algorithm for reconstructing, from 1 K set of valuations F ” of the base field “ ”, hence also the the algebraic closure “ F — cf., e.g., [AbsTopIII], Corollary F )” [e.g., as a collection of topologies on V ( “ arc V ( K ) ∈ , let us recall [cf. Remark 3.1.2; [AbsTopIII], w 2.8]. Moreover, for  D ( ), the Corollaries 2.8, 2.9] that one may π ,from reconstruct group-theoretically 1  † associated to .Let D be a category equivalent C C Aut -holomorphic orbispace w w  to D . Then let us write †  ( ) V D †  ( V for the set of valuations [i.e., “ D ( F )”], equipped with its natural π )-action, 1 def    † † † V ( ( V ) /π D D = D ) ) ( 1 arc  † †   † for the quotient of ( D )by ) [i.e., “ V ( K )”], and, for w ∈ V ( D D π ) V , ( 1  † C ,w D ( ) C [i.e., “ ” — cf. the discussion of [AbsTopIII], Definition 5.1, (ii)] for the Aut- w holomorphic orbispace obtained by applying these group-theoretic reconstruction †  algorithms to π D U is an arbitrary Aut -holomorphic orbispace ,then ( ). Now if 1 morphism let us define a  † D → U to be a morphism of Aut-holomorphic orbispaces [cf. [AbsTopIII], Definition 2.1,  †  † arc w D ) ,w )forsome . Thus, it makes sense to speak of the ∈ V ( ( D C U → (ii)]  † with D → pre-composite (respectively, post-composite) of such a morphism U a morphism of Aut-holomorphic orbispaces (respectively, with an isomorphism [cf. ∼ ‡   ‡ †   [i.e., where → D D is a category equivalent to D ]). Finally, just as in D 0] § ⋂ good non V ∈ V ”, we may apply [AbsTopI], the discussion of (ii) in the case of “ v Lemma 4.5 [cf. also Remark 1.2.2, (ii), of the present paper], to conclude that it  † set of label classes ,aswellasthe D of set of cusps makes sense to speak of the of cusps †  D ) LabCusp(  †  F -torsor structure . D , which admits a natural of l †   † † D , be a category equivalent to D = { D D -prime- } D a (vi) Let V ∈ v v  † † to be a collection of V , then we define a poly-morphism ∈ D D → strip v .If v non arc † †  D D [cf. § 0when v ∈ V ]. We define a → v ∈ V ;(v)when morphisms v poly-morphism † †  D → D †  e † is a → to be a collection of poly-morphisms D { } } D D . Finally, if { V e ∈ E v ∈ v D -prime-strips , then we define a poly-morphism capsule of e † †  e { D → } D D (respectively, { ) D } → ∈ E e E e ∈

98 98 SHINICHI MOCHIZUKI † e † e  } D { to be a collection of poly-morphisms (respectively, { D D → → D } ). ∈ e e ∈ E E The following result follows immediately from the discussion of Definition 4.1, (ii). Proposition 4.2. (The Set of Label Classes of Cusps of a Base-Prime- † † = { ,there D V } ∈ ,w v be a D D . Then for any -prime-strip Let Strip) v V v ∈ exist bijections ∼ † † → LabCusp( D D ) ) LabCusp( w v by the condition that they be that are with uniquely determined compatible  † † →  η [cf. Definition 4.1, (ii)], as well as with the F - η the assignments l v w on either side. In particular, these bijections are torsor structures preserved of -prime-strips. Thus, by identifying the various isomorphisms D by arbitrary † † ) ” via these bijections, it makes sense to write LabCusp( D D . ) LabCusp( “ v † † D ) is equipped with a canonical element , arising from the η Finally, LabCusp( v  -torsor structure ], as well as a natural F v [for V ; in particular, this canonical ∈ l  element and F -torsor structure determine a natural bijection l ∼  † D ) → F LabCusp( l that is preserved by isomorphisms of D -prime-strips. good — V ∈ v Note that if, in Examples 3.3, 3.4 — i.e., at Remark 4.2.1. ” ”bymeansof“ ” instead of “ X exist not C , then there does D one defines “ v v → − v a system of bijections as in Proposition 4.2. Indeed, by the Tchebotarev density theorem 4, Theorem 10], it follows immediately that [cf., e.g., [Lang], Chapter VIII, § → V such that, for a suitable embedding Gal( K/F ) ↪ ∈ GL ( F ), the there exist v 2 l ( F ) determined [up to conjugation] K/F ) decomposition subgroup in Gal( → ↪ GL l 2 is equal to the subgroup of diagonal matrices with determinant 1. Thus, if by v † † † † D } , v D = { { D ,the } D are as in Definition 4.1, (i), then for such a = v ∈ v V V v ∈ v † acts transitively on the set of label classes of cusps of D automorphism group of v † † , while the automorphism group of D D acts trivially [by [EtTh], Corollary 2.9] w v bad † w ∈ V D . for any on the set of label classes of cusps of w In the following, we construct the Example 4.3. Model Base-NF-Bridges. for the notion of a “base-NF-bridge” [cf. Definition 4.6, (i), below]. “models” (i) Write  ∼ ∼ ) D ) ⊆ Aut( C Aut( ) C ( Out(Π ) Aut = = C  K K K ∼ — where the first “ ” follows, for instance, from [AbsTopIII], Theorem 1.9 — for = the subgroup of elements which fix the cusp  . Now let us recall that the profi- may be reconstructed group-theoretically from Π [cf. [AbsTopII], nite group Δ X C K Corollary 3.3, (i), (ii); [AbsTopII], Remark 3.3.2; [AbsTopI], Example 4.8]. Since

99 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 99 by multiplication by 1onthe l -torsion inner automorphisms of Π clearly act ± C K ab points of E ⊗ F ], we obtain a natural homomorphism Out(Π [i.e., on Δ ) → l C X F K ab F ⊗ / {± 1 } . Thus, it follows immediately from the discussion of the nota- ) Aut(Δ l X ”, and “  ” in Definition 3.1, (c), (d), (f) [cf. also Remark 3.1.5; the ”, “ C tion “ K K discussion preceding [EtTh], Definition 2.1; the discussion of [EtTh], Remark 2.6.1], ∼ ab F arising from ) / {± 1 } ⊗ → GL } ( F 1 ) / {± that, relative to an isomorphism Aut(Δ l l 2 X ab GL } ⊗ F 1 ,ifwewriteIm( G {± / ) ⊆ for ) ( F for Δ a suitable choice of basis F l l 2 mod X def )on = Gal( F/F {± 1 } ]of G the image of the natural action [i.e., modulo mod F mod [cf. the homomorphism of the display of Definition 3.1, E the -torsion points l of F ” discussed in Remark 3.1.7], then the images of the groups C (c); the model “ F mod ( C ) may be identified with the subgroups consisting of elements ), Aut( C Aut  K K of the form )} )} {( {( ( ) ∗∗ ∗∗ {± / 1 ) F ( SL ⊇ } ) G Im( ⊆ ⊆ l 2 F mod ∗ 0 ± 1 0 ⊆ ) 1 ” and “Borel” subgroups — of Im( G — i.e., “semi-unipotent, up to ± F mod {± ( F .Write ) / } 1 GL l 2 SL SL Aut ( C C ) ⊆ Aut Aut( ( C ⊆ ) , Aut ) ( C )   K K K K act trivially on the subfield ( μ for the respective subgroups of elements that F ⊆ K ) l [cf. Remark 3.1.7, (iii)] and def def un Bor ± ) ) ) · V =Aut V ( C =Aut( C K ( · V ⊆ V ⊆ V  K K for the resulting subsets of V ( K ). Thus, one verifies immediately that the subgroup , and that we have ( C normal ) ⊆ Aut( natural isomorphisms )is C Aut  K K ∼ ∼  SL SL Aut Aut C F ( C → ) / → Aut( C ) ) / Aut ) ( C (   K K K K l Bor ± un  —sowemaythinkof V -orbit of V F as the . Also, we observe that [in light l of the above discussion] it follows immediately that there exists a group-theoretic  ( D ) [i.e., an isomorph of Π ] the subgroup algorithm for reconstructing, from π C 1 K   Aut ) D Aut( D ( ) ⊆  ). ( C determined by Aut  K non (ii) Let v . Then the natural restriction functor on finite ́ V ∈ etale coverings good v → V C ∈ → C if X arising from the natural composite morphism K v → − v bad X ) determines [cf. Examples 3.2, (i); → C (respectively, → C V if v ∈ K v v NF  3.3, (i)] a φ natural morphism : D →D 0 for the definition of the term § [cf. v ,v • “morphism”]. Write  NF →D : D φ v v  poly-morphism D →D for the of the form given by the collection of morphisms v NF ◦ φ β ◦ α • ,v

100 100 SHINICHI MOCHIZUKI ∼ ∼ Aut( X X Aut( α D ∈ ∈ β )); ) (respectively, α ∈ Aut( D ) ) —where Aut( = = v v − → v v  ∼ Aut D ) Aut ) [cf., e.g., [AbsTopIII], Theorem 1.9]. ( C ( =   K arc ∈ . Thus, [cf. Example 3.4, (i)] we have a tautological morphism V (iii) Let v ∼   NF D → C D X → C [cf. Definition D = ,v ), hence a morphism φ →D : ( v v ,v • v → − v 4.1, (v)]. Write NF  φ : →D D v v  given by the collection of morphisms D for the →D poly-morphism of the form v NF β ◦ φ ◦ α • ,v  ∼ ∼ ( D Aut Aut( X ∈ β ) ) [cf. [AbsTopIII], Corollary 2.3]; ) D Aut( ∈ α —where = =  v − → v Aut C ( ).  K  ,let F (iv) For each j ∈ l = {D } D j v v ∈ V j — where we use the notation v to denote the pair ( j, v )—bea copy of the j “tautological -prime-strip” {D D } . Let us denote by v v V ∈ NF  φ : D →D 1 1  F [where, by abuse of notation, we write “1” for the element of determined by 1] l NF  determined by the collection poly-morphism the { φ of copies of →D D } : v v ∈ V v 1 1 NF NF φ the poly-morphisms constructed in (ii), (iii). Note that φ is stabilized by the v 1   Aut action of ) , ( D C . Thus, it makes sense to consider, for arbitrary j ∈ F on  K l the poly-morphism NF  : D →D φ j j ∼ D obtained [via any isomorphism ]by post-composing with the “poly-action” D = j 1   on D .Letuswrite of [i.e., action via poly-automorphisms — cf. (i)] F ∈ j l def  D D } = {  j F j ∈ l  D -prime-strips indexed by j ∈ F for the capsule of [cf. Definition 4.1, (iv)] and l denote by NF  : D →D φ   NF  given by the collection of poly-morphisms { } φ .Thus, poly-morphism the j ∈ j F l  NF  φ with respect to the natural poly-action of F is and the on D equivariant  l  0] poly-automorphisms, , via capsule-full [cf. § of F natural permutation poly-action l . In particular, we obtain a natural poly-action D on the constituents of the capsule   NF  ,φ ). D D , on the collection of data ( F of   l

101 ̈ INTER-UNIVERSAL TEICHM 101 ULLER THEORY I Remark 4.3.1. . Note (i) Suppose, for simplicity, in the following discussion that F F = mod K → Spec( F ) [or, equivalently, the homomor- that the morphism of schemes Spec( ) → K ] does not admit a section . This nonexistence of a section phism of rings F↪ of the is closely related to the nonexistence of a “global multiplicative subspace” , this nonexis- sort discussed in [HASurII], Remark 3.7. In the context of loc. cit. tence of a “global multiplicative subspace” may be thought of as a concrete way principal obstruction to applying the scheme-theoretic Hodge- of representing the Arakelov theory of [HASurI], [HASurII] to diophantine geometry. From this point F , of view, if one thinks of the ring structure of as a sort of “arithmetic holo- K morphic structure” [cf. [AbsTopIII], Remark 5.10.2, (ii)], then one may think of [ -]prime-strips that appear in the discussion of Example 4.3 as defining, via the D NF of Example 4.3, (iv), the arrows φ j of Spec( K ) → “arithmetic collections of local analytic sections” F ) Spec( — cf. Fig. 4.1 below, where each “ ... −·−· ” represents a [ D ·−·− -]prime-strip. In fact, if, for the sake of brevity, we abbreviate the phrase “collection of local an- alytic” by the term “local-analytic” , then each of these sections may be thought of as yielding not only an “arithmetic local-analytic global multiplicative sub- space” , but also an “arithmetic local-analytic global canonical generator” [i.e., up to multiplication by ± -torsion points of 1, of the quotient of the module of l the elliptic curve in question by the “arithmetic local-analytic global multiplicative subspace”]. We refer to Remark 4.9.1, (i), below, for more on this point of view. ... −·−·−· ·−·−·−  Gal( K/F ) −·−·−· ... ·−·−·− ... K ⊆ GL F ( ) l 2 ·−·−·− ... −·−·−· −·−·−· ... ·−·−·− ⏐ ⏐  ... F ·−·−·− −·−·−· K ) → Fig. 4.1: Prime-strips as “sections” of Spec( F ) Spec( (ii) The way in which these “arithmetic local-analytic sections” constituted by the [ D -]prime-strips fail to be [globally] “arithmetically holomorphic” may be understood from several closely related points of view. The first point of view was already noted above in (i) — namely: (a) these sections fail to extend to ring homomorphisms K → F . The second point of view involves the classical phenomenon of decomposition of primes in extensions of number fields. The decomposition of primes in extensions

102 102 SHINICHI MOCHIZUKI tree , as in Fig. 4.2, below. If one thinks of number fields may be represented by a of the tree in large parentheses of Fig. 4.2 as representing the decomposition of v in extensions of F [such as K !], then the “arithmetic of primes over a prime F D local-analytic sections” constituted by the -prime-strips may be thought of as (b) an isomorphism, or identification, between v [i.e., a prime of F ]and ′ K [i.e., a prime of ] which [manifestly — cf., e.g., [NSW], Theorem v isomorphism between the respective prime 12.2.5] fails to extend to an ′ . decomposition trees v over and v ∈ ” between sets in axiomatic set theory as determining If one thinks of the relation “ “tree” ,then a reminiscent the point of view of (b) is § 3, of the point of view of [IUTchIV], constructing some sort of artificial solution to where one is concerned with ∈ a ” [cf. the discussion of [IUTchIV], Remark the “membership equation a 3.3.1, (i)]. The third point of view consists of the observation that although the “arithmetic D -prime-strips involve isomorphisms of local-analytic sections” constituted by the the various local absolute Galois groups , (c) these isomorphisms of local absolute Galois groups fail to extend to a  G [i.e., a section of the section of global absolute Galois groups G K F ]. → ↪ G natural inclusion G K F Here, we note that in fact, by the Neukirch-Uchida theorem [cf. [NSW], Chapter XII, § 2], one may think of (a) and (c) as essentially equivalent . Moreover, (b) is closely related to this equivalence, in the sense that the proof [cf., e.g., [NSW], 2] of the Neukirch-Uchida theorem depends in an essential fashion Chapter XII, § of the number fields involved. on a careful analysis of the prime decomposition trees ⎞ ⎛ ⎛ ⎞ ... ... ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ \| / ... ... ⎜ ⎟ ⎟ ⎜ \| / ⎝ ⎠ ⎟ ⎜ ⎟ ⎜ ′′ ′′′ ′ ⊇ v v v ′ ⎟ ⎜ v ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ \| / ⎠ ⎝ v Fig. 4.2: Prime decomposition trees (iii) In some sense, understanding more precisely the content of the failure of these “arithmetic local-analytic sections” constituted by the D -prime-strips to be “arithmetically holomorphic” is a central theme of the theory of the present series of papers — a theme which is very much in line with the spirit of classical complex Teichm ̈ uller theory . Remark 4.3.2. The incompatibility of the “arithmetic local-analytic sections” of Remark 4.3.1, (i), with global prime distributions and global absolute Galois groups [cf. the discussion of Remark 4.3.1, (ii)] is precisely the technical obstacle that

103 ̈ INTER-UNIVERSAL TEICHM 103 ULLER THEORY I absolute p will necessitate the application — in [IUTchIII] — of the -adic mono- “panalocalization along developed in [AbsTopIII], in the form of anabelian geometry [cf. [IUTchIII] for more details]. Indeed, the various prime-strips” the mono-anabelian theory developed in [AbsTopIII] represents the cul- mination of earlier research of the author during the years 2000 to 2007 concerning -adic anabelian geometry — research that was absolute p developing a geometry that would allow motivated precisely by the goal of one to work with the “arithmetic local-analytic sections” constituted by the prime-strips, so as to overcome the principal technical obstruction to applying the Hodge-Arakelov theory of [HASurI], [HASurII] [cf. Remark 4.3.1, (i)]. Note that the “desired geometry” in question will also be subject to other require- ments. For instance, in [IUTchIII] [cf. also [IUTchII], 4], we shall make essential § use of the global arithmetic — i.e., the ring structure and absolute Galois groups — of number fields . As observed above in Remark 4.3.1, (ii), these global arithmetic not compatible with the “arithmetic local-analytic sections” consti- structures are tuted by the prime-strips. In particular, this state of affairs imposes the further requirement that the “geometry” in question be compatible with globalization , i.e., that it give rise to the global arithmetic of the number fields in question in a fashion that is independent of the various local geometries that appear in the “arithmetic local-analytic sections” constituted by the prime-strips, but nevertheless admits lo- calization operations to these various local geometries [cf. Fig. 4.3; the discussion of [IUTchII], Remark 4.11.2, (iii); [AbsTopIII], Remark 3.7.6, (iii), (v)]. local geometry local geometry local geometry ... ... ′′ ′ v at at v at v ↖↑↗ global geometry Fig. 4.3: Globalizability Finally, in order for the “desired geometry” to be applicable to the theory developed in the present series of papers, it is necessary for it to be based on “ ́ etale-like structures” ,soastogiveriseto canonical splittings ,asinthe ́ etale-picture discussed in Corollary 3.9, (i). Thus, in summary, the requirements that we wish to impose on the “desired geometry” are the following: (a) local independence of global structures , (b) globalizability , in a fashion that is independent of local structures , (c) the property of being based on ́ etale-like structures . Note, in particular, that properties (a), (b) at first glance almost appear to con- tradict one another . In particular, the simultaneous realization of (a), (b) is highly

104 104 SHINICHI MOCHIZUKI . For instance, in the case of a function field of dimension one over a nontrivial base field, the simultaneous realization of properties (a), (b) appears to require descend to the that one restrict oneself essentially to working with structures that of the theory of [AbsTopIII] !Itisthusa base field highly nontrivial consequence of [AbsTopIII] does indeed satisfy all of these that the mono-anabelian geometry requirements (a), (b), (c) [cf. the discussion of [AbsTopIII], I1]. § Remark 4.3.3. mono- (i) One important theme of [AbsTopIII] is the analogy between the of [AbsTopIII] and the theory of Frobenius-invariant indige- anabelian theory p nous bundles uller theory [cf. [AbsTopIII], of the sort that appear in -adic Teichm ̈ I5]. In fact, [although this point of view is not mentioned in [AbsTopIII]] one may § p “compose” -adic and complex theo- this analogy with the analogy between the Ord], Introduction; [ p ries discussed in [ § 0, and consider the p Teich], Introduction, classical ge- analogy between the mono-anabelian theory of [AbsTopIII] and the H . In addition to being more elementary ometry of the upper half-plane than the -adic theory, this analogy with the classical geometry of the upper half-plane p also has the virtue that H — i.e., the Poin- ahler metric canonical K ̈ since it revolves around the car ́ emetric — on the upper half-plane, it renders more transparent the relationship between the theory of the present series of papers and clas- sical Arakelov theory [which also revolves, to a substantial extent, around ahler metrics at the archimedean primes]. K ̈ (ii) The essential content of the mono-anabelian theory of [AbsTopIII] may be summarized by the diagram × log ) k −→ k  Π( ∗  Π k is an algebraic closure of k ; Π is the arith- ; k is a finite extension of Q —where p metic fundamental group of a hyperbolic orbicurve over k ; log is the p -adic loga- ) denotes the “tautological rithm [cf. [AbsTopIII], , ∇ I1]. On the other hand, if ( § E E indigenous bundle” [i.e., the first de Rham cohomology of the tautological on H →E→ ], then one has a natural 0 H ω Hodge filtration τ → 0 elliptic curve over → def − 1 = ω are holomorphic line bundles on H ], together with a natural , ω [where τ : E→E . The composite ι complex conjugation operation E ι E ω↪ →E −→ E  τ on ω . For any trivializing section f of then determines an |−| Hermitian metric ω ,the(1 , 1)-form ω 1 def 2 ∂ log( | f ) ∂ = | κ H ω πi 2 is the canonical K ̈ ahler metric [i.e., Poincar ́ e metric] on H . Then one can already and the diagram ( ∗ ) reviewed above: identify various formal similarities between κ H Indeed, at a purely formal [but by no means coincidental!] level, the “log” that

105 ̈ INTER-UNIVERSAL TEICHM 105 ULLER THEORY I κ appears in the definition of log . is reminiscent of the “log-Frobenius operation” H At a less formal level, the “Galois group” Π is reminiscent — cf. the point of view that , a point of view that underlies “Galois groups are arithmetic tangent bundles” the theory of the arithmetic Kodaira-Spencer morphism discussed in [HASurI]! — . If one thinks of complex conjugation as a sort of “archimedean Frobenius” [cf. of ∂ ∂ is reminiscent of the “Galois group” Π operating p Teich], Introduction, 0], then [ § ] of the log-Frobenius operation log . The Hodge filtration opposite side ι on the [cf. E log on either side of k [cf. ring structures of the copies of E of corresponds to the the discussion of [AbsTopIII], Remark 3.7.2]. Finally, perhaps most importantly from the point of view of the theory of the present series of papers: play the role in the theory of [AbsTopIII] of log-shells “canon- the fact that § I1] — i.e., ical rigid integral structures” [cf. [AbsTopIII], “canonical stan- — is reminiscent of the fact that the K ̈ ahler metric dard units of volume” determining a canonical notion of volume also plays the role of . on H κ H (iii) From the point of view of the analogy discussed in (ii), property (a) of Remark 4.3.2 may be thought of as corresponding to the local representabil- —of —on,say,acompactquotient S of H ity via the [positive] (1 , 1) -form κ H the [positive] global degree of [the result of descending to S ] the line bundle ω ; property (b) of Remark 4.3.2 may be thought of as corresponding to the fact that that gives rise to a local representation on S of the notion of this (1 κ , 1)-form H S canonical a positive global degree not only exists locally on , but also admits a S which may be related to the global extension to the entire Riemann surface S ]. [i.e., of algebraic rational functions on algebraic theory (iv) The analogy discussed in (ii) may be summarized as follows: geometry of the upper-half plane H mono-anabelian theory the differential operator ∂ the Galois group Π the Galois group Π the differential operator on the opposite side of log ∂ the Hodge filtration of E , the ring structures of the copies k on either side of log ι of , |−| E E log-shells as the ahler volume canonical K ̈ κ canonical units of volume H Θ -Bridges. Example 4.4. Model Base- In the following, we construct the “models” for the notion of a “base-Θ-bridge” [cf. Definition 4.6, (ii), below]. We continue to use the notation of Example 4.3. bad natural bijection ∈ V . Recall that there is a between the set v (i) Let ⋃ def  | F 1 = {± F } =0 | F / l l l

106 106 SHINICHI MOCHIZUKI {± [i.e., the set of -orbits of F 1 ] and the set of cusps of the hyperbolic orbicurve } l [cf. [EtTh], Corollary 2.9]. Thus, [by considering fibers over ] we obtain C C v v X of various collections of cusps of .Write X | , F ∈| labels l v v ∈ X ) ( K μ v − v unique torsion point of order X for the 2 whose closure in any stable model of v intersects the same irreducible component of the special fiber of the stable over O K v | . Now observe that it makes sense to model as the [unique] cusp labeled 0 ∈| F l ,relativetothe -translates of the cusps μ ( K ) obtained as X ∈ speak of the points v − v [i.e., whose origin group scheme structure of the elliptic curve determined by X v | ]. We shall refer to these μ -translates of the is given by the cusp labeled 0 F ∈| l − as the | evaluation points of X . Note that the value of cusps with labels F ∈| l v ” of Example 3.2, (ii), at a point lying over an evaluation the theta function “Θ v j F point arising from a cusp with label ∈| | -orbit of is contained in the μ l 2 l 2 j { q } j ≡ j v j [cf. Example 3.2, (iv); [EtTh], Proposition 1.4, (ii)] — where ranges over the Z that map to j elements of F ∈| | . In particular, it follows immediately from the l -th roots of the theta l [i.e., by considering X → of the covering definition X v v X function! — cf. [EtTh], Definition 2.5, (i)] that the points of that lie over v evaluation points of X are all defined over K . We shall refer to the points v v of ( X evaluation points ) that lie over the evaluation points of X as the K X ∈ v v v v and to the various sections tp G → =Π Π v v X v  G of the natural surjection Π that arise from the evaluation points as the v v  G . Thus, each evaluation section has an associ- of Π evaluation sections v v | . Note that there is a group-theoretic algorithm for constructing ated F ∈| label l . Indeed, this topological group Π the evaluation sections from [isomorphs of] the v follows immediately from [the proofs of] [EtTh], Corollary 2.9 [concerning the group- theoreticity of the labels ]; [EtTh], Proposition 2.4 [concerning the group-theoreticity ]; [SemiAnbd], Corollary 3.11 [concerning the dual semi-graphs of the , Π of Π X C v v tp tp special fibers of stable models], applied to Δ ⊆ Π =Π ;[SemiAnbd],The- v X X v v group-theoreticity of the decomposition groups of orem 6.8, (iii) [concerning the ]. -translates of the cusps μ − bad v (ii) We continue to suppose that ∈ V .Let D } = {D >,w w ∈ V >  copy of the “tautological D -prime-strip” {D ,write } .Foreach F ∈ be a j V w w ∈ l Θ →D : D φ >,v v v j j

107 ̈ INTER-UNIVERSAL TEICHM 107 ULLER THEORY I § poly-morphism given by the collection of morphisms [cf. 0] obtained by for the ∼ ∼ 0 0 temp temp composing with arbitrary isomorphisms D ) B (Π (Π →B ) →D , v >,v v v j temp 0 temp 0 ) B (Π →B ) (Π that arise [i.e., via composition the various morphisms v v ]fromthe G evaluation sections labeled j .Now  with the natural surjection Π v v temp 0 B is any isomorph of C if ) , then let us write (Π v geo ( C ) ⊆ π ) ( C π 1 1 tp tp , a subgroup which we re- ⊆ Π =Π for the subgroup corresponding to Δ v X X v v [cf., e.g., [AbsTopI], Theorem 2.6, reconstructed group-theoretically call may be (v); [AbsTopI], Proposition 4.10, (i)]. Then we observe that for each constituent Θ φ of the poly-morphism →D , the induced homomorphism morphism D >,v v v j j π D ) [well-defined, up to composition with an inner automorphism] D ) → π ( ( 1 >,v v 1 j is [ofthedomainandcodomainofthis compatible with the respective outer actions geo geo ( ) for some [not necessarily unique, but D ( ), D π homomorphism] on π v >,v 1 1 j finite ambiguity determined up to — cf. [SemiAnbd], Theorem 6.4!] outer isomor- ∼ geo geo Θ D φ “ ) D → π is ). We shall refer to this fact by saying that ( ( π phism >,v v v 1 1 j j compatible with the outer actions on the respective geometric [tempered] fundamen- . tal groups” good  (iii) Let v ∈ .Foreach j ∈ F V ,write l ∼ Θ φ D →D : v >,v v j j § 0]. for the full poly-isomorphism [cf.  ,write F ∈ j (iv) For each l Θ φ D → D : > j j Θ { } for the poly-morphism and : D determined by the collection →D φ v >,v V v ∈ v j j Θ : D → D φ >   Θ  φ } for the poly-morphism nat- admits a D { . Thus, whereas the capsule  j ∈ j F l  ural permutation poly-action by F ,the “labels” — i.e., in effect, elements of l ) [cf. Proposition 4.2] — determined by the various collections of LabCusp( D >  held fixed by arbitrary are evaluation sections ∈ F j corresponding to a given l [cf. Proposition 4.2]. automorphisms of D > Example 4.5. Transport of Label Classes of Cusps via Model Base- We continue to use the notation of Examples 4.3, 4.4. Bridges.  j F (i) Let ∈ , . Recall from Example 4.3, (iv), that the data of the ∈ V v l non  NF NF  arrow φ →D : at v consists of an arrow φ ,then V ∈ : D v .If →D D j v v j j j  NF D ( induces various outer homomorphisms π ); thus, ( D π → ) φ v 1 1 v j j

108 108 SHINICHI MOCHIZUKI  π ) whose unique index l cuspidal inertia groups of D by considering ( 1 contained in the image subgroup is of this homomorphism [cf. Corollary bad ∈ V ; the discussion of Remark 4.5.1 below], v 2.5 when  -torsors F natural isomorphism of we conclude that these homomorphisms induce a l ∼ arc  D LabCusp( → ). In a similar vein, if v ∈ V LabCusp( , then it follows from D ) v j NF Definition 4.1, (v), that φ consists of certain morphisms of Aut-holomorphic v j  )from D ( D ) → π ( π orbispaces which induce various outer homomorphisms 1 1 v j  π the [discrete] topological fundamental group D ) to the profinite group π ); ( D ( 1 1 v j thus,  by considering the closures in π D of the images of cuspidal inertia ( ) 1 ( ) [cf. the discussion of Remark 4.5.1 below], D π groups of 1 v j  F we conclude that these homomorphisms induce a natural isomorphism of -torsors l ∼  LabCusp( D D ). Now let us observe that it follows immediately LabCusp( ) → v j  from the definitions that, as one allows v vary to -torsors F , these isomorphisms of l ∼  LabCusp( D LabCusp( D ) → ) are compatible with the natural bijections in the v j  first display of Proposition 4.2, hence determine an isomorphism of F -torsors l ∼  D LabCusp( → LabCusp( D ). Next, let us note that the data of the arrow ) j  Θ D -torsors → D ∈ at the various v : V determines an isomorphism of F φ > j j l ∼ )[whichmaybe ) → LabCusp( D with the previous isomor- composed D LabCusp( > j ∼   )]. Indeed, this is immediate D ) -torsors LabCusp( → LabCusp( D F phism of j l bad good ∈ v ∈ V , it follows immediately from ;when V from the definitions when v the discussion of Example 4.4, (ii). (ii) The discussion of (i) may be summarized as follows:  NF Θ F ∈ j for each v ∈ V via φ , restriction , φ at the various determines j j l  an isomorphism of F -torsors l ∼ LC  φ D : LabCusp( → LabCusp( D ) ) > j LC LC is obtained from φ by composing with the action by φ such that 1 j  F j ∈ . l  Write [  ) for the element determined by D ] ∈  . Then we observe that LabCusp( LC LC → ([  ])  j j ; φ →  ( j · [  ]) φ 1 j ∼  LabCusp( via the natural bijection D F of Proposition 4.2. In particular, ) → > l  LabCusp( ] ∈ )maybe characterized as the unique element η ∈ D the element [   LC φ ) such that evaluation at η yields the assignment . j →  LabCusp( D j Remark 4.5.1. G be a group. If H ⊆ G is a subgroup, g ∈ G , then we shall write (i) Let def g 1 − .Let g H · g = G · J ⊆ H ⊆ be subgroups. Suppose further that each of the H

109 ̈ INTER-UNIVERSAL TEICHM 109 ULLER THEORY I J subgroups of G is only known up to conjugacy in G . Put another way, we , H G independent -conjugacy suppose that we are in a situation in which there are and . Thus, for instance, indeterminacies H J in the specification of the subgroups ι : there is → H from its γ - no natural way to distinguish the given inclusion J↪ γ γ γ J . Moreover, it may happen to be the case that → H : ,for γ ∈ G ↪ conjugate ι − 1 g g , but also J for some ∈ G , not only g J H [or, equivalently J H ]. Here, the ⊆ ⊆ g are ; indeed, the abstract pairs of H H not necessarily conjugate in J J subgroups , g need not be isomorphic ) H,J )and( of a group and a subgroup given by ( H,J H [i.e., it is not even necessarily the case that there exists an automorphism of g ]. In particular, the existence of the independent G -conjugacy onto J J that maps and means that one cannot specify the J indeterminacies in the specification of H − 1 g H independently of the inclusion ζ : J↪ → H ι : inclusion → J↪ [i.e., arising from out g J ⊆ ” ↪ → ]. One way to express this state of affairs is as follows. Write “ H for the outer homomorphism determined by an injective homomorphism between out out → ↪ → H ↪ G of the natural J groups. Then the collection of factorizations out J “outer” inclusion → G through some ↪ -conjugate of H — i.e., put another G way, outer homomorphisms the collection of out J → H ↪ out compatible with the “structure morphisms” J that are ↪ → G , out H → determined by the natural inclusions ↪ G compatible with independent ,inafashionthatis G well-defined —is -conjugacy . That is to say, this collection and J indeterminacies in the specification of H g g 2 1 J of outer homomorphisms amounts to the collection of inclusions → H ↪ ,for ,g [together with, say, ∈ G . By contrast, to specify the inclusion ι : J↪ → H g 2 1 − 1 g γ : ] } of the inclusion ζ G J↪ → H - [and its independently G its { ι -conjugates G γ ∈ γ partial synchronization } — ] amounts to the imposition of a ζ conjugates { ∈ γ G i.e., a partial deactivation — of the [a priori!] independent G -conjugacy indeter- minacies in the specification of J and H . Moreover, such a “partial deactivation” can only be effected at the cost of introducing certain arbitrary choices into the construction under consideration. factorizations (ii) Relative to the considered in (i), we make the following ∗ ∗ , the condition of H and a subgroup H ⊆ I H -conjugate G observation. Given a on I that ⊆ ( ∗ I G -conjugate of J ) be a ∗ that , while the condition on I of the datum independent H is a condition that is ∼ ∗ = ∼ ( ∗ ) be a J such that ( H I ,I -conjugate of ) G ( H,J ) = ∼ ” denotes an isomorphism of pairs consisting of a group and a sub- [where the “ = group — cf. the discussion of (i)] is a condition that depends , in an essential fashion, ⊆ ∗ H on the datum ∗ ) is precisely the condition that one must impose when . Here, ( ∼ = ) is the condition that one ∗ one considers arbitrary factorizations as in (i), while ( must impose when one wishes to restrict one’s attention to factorizations whose

110 110 SHINICHI MOCHIZUKI .Thatisto first arrow gives rise to a pair isomorphic to the pair determined by ι ∼ ∗ = ) on the datum H may be regarded as an explicit formu- of ( ∗ dependence say, the as discussed lation of the necessity for the “imposition of a partial synchronization” ∗ ⊆ ), of the datum H independence ∗ in (i), while the corresponding , exhibited by ( of such a necessity when one lack may be regarded as an explicit formulation of the considers arbitrary factorizations as in (i). Finally, we note that by reversing the ”, one may consider a subgroup I ⊆ G that direction of the inclusion “ a ⊆ contains ∗ ∗ of J , i.e., I ⊇ J ; then analogous observations may be made J G given -conjugate ⊇ -conjugate of I that I be a G )on H . concerning the condition ( ∗ (iii) The abstract situation described in (i) occurs in the discussion of Example bad ∈ V ” (respectively, “ . That is to say, the group “ G ”) of (i) H ”; “ J v 4.5, (i), at   D ( ) (respectively, the image of π ); the ( D ( D )in π corresponds to the group π 1 v 1 1 j  l π unique index open subgroup of a cuspidal inertia group of ( )) of Example 4.5, D 1  D ( ) is only known up to D ( ) → π π (i). Here, we recall that the homomorphism 1 1 v j  composition with an inner automorphism — i.e., up to π ; a cuspi- ) -conjugacy ( D 1   ( D ) is also only determined by an element ∈ LabCusp( D ) dal inertia group of π 1  -conjugacy ) ( D . Moreover, it is immediate from the construction of the π up to 1 -NF-bridges” D of Example 4.3 [cf. also Definition 4.6, (i), below] that “model there synchronize these indeterminacies way to is no natural . Indeed, from the point of view of the discussion of Remark 4.3.1, (ii), by considering the actions of the absolute Galois groups of the local and global base fields involved on the cuspi- dal inertia groups that appear, one sees that such a synchronization would amount, Galois-equivariant splitting roughly speaking, to a [i.e., relative to the global abso- lute Galois groups that appear] of the “prime decomposition trees” of Remark 4.3.1, (ii) — which is absurd [cf. [IUTchII], Remark 2.5.2, (iii), for a more detailed discus- sion of this sort of phenomenon]. This phenomenon of the “non-synchronizability” of indeterminacies arising from local and global absolute Galois groups is reminis- cent of the discussion of [EtTh], Remark 2.16.2. On the other hand, by Corollary fact that 2.5, one concludes in the present situation the highly nontrivial “ J↪ → H↪ a G ”is uniquely determined by the com- factorization → J↪ → G , i.e., by the G -conjugate of J that one starts with, without posite a priori resorting to any “synchronization of indeterminacies” . (iv) A similar situation to the situation of (iii) occurs in the discussion of arc . Thatistosay,inthiscase,thegroup“ G ”(re- V ∈ Example 4.5, (i), at v  ( D ) (respectively, the H spectively, “ J ”) of (i) corresponds to the group π ”; “ 1  π D D ( )in ( π ( D )) of Example 4.5, ); a cuspidal inertia group of image of π 1 1 v v 1 j j factorization “ J↪ → H↪ → G ”is (i). In this case, although it does not hold that a G uniquely determined → G , i.e., by the J↪ -conjugate of J that by the composite one starts with [cf. Remark 2.6.1], it does nevertheless hold, by Corollary 2.8, that the H -conjugacy class of the image of J via the arrow J↪ → H that occurs in such a factorization is uniquely determined . arc ∈ V in (iv) is somewhat than the weaker v (v) The property observed at bad ∈ V in (iii). In the present series of pa- property observed at v rather strong pers, however, we shall only be concerned with such subtle factorization proper- bad ∈ V , where we wish to develop, in [IUTchII], the theory of “Hodge- ties at v

111 ̈ INTER-UNIVERSAL TEICHM 111 ULLER THEORY I by restricting certain cohomology classes via an ar- Arakelov-theoretic evaluation” ” appearing in a ” of the sort discussed “ J↪ → H↪ → G H factorization row “ J↪ → in (i). In fact, in the context of the theory of Hodge-Arakelov-theoretic evaluation that will be developed in [IUTchII], a slightly modified version of the phenome- version to be developed in “additive” non discussed in (iii) — which involves the 4 — will be of central § 6 of the “multiplicative” theory developed in the present § importance. Definition 4.6. (i) We define a -NF-bridge , [relative to the given initial base-NF-bridge ,or D Θ-data] to be a poly-morphism † NF φ  †  † D D −→ J †  † †  } D D = { is a D category equivalent to ; is a capsule of D - —where D J j ∈ J j J — such that there exist isomorphisms , indexed by a finite index set prime-strips ∼ ∼ † † NF  NF  † → . We define a(n) D , , conjugation by which maps φ D D  → → φ D  J   -NF-bridges [iso]morphism of D ‡ † NF NF φ φ   † ‡  †  ‡ ′ ( → ) D −→ −→ ( D D ) D J J to be a pair of poly-morphisms ∼ ∼  ‡ †  † ‡ ′ D D D D → ; → J J ∼ † ‡  †  ‡ ′ → is a —where D § 0]; is a D D → capsule-full poly-isomorphism D [cf. J J  ‡  † D ( )- [or, equivalently, Aut D ( )-] orbit [cf. the poly-morphism which is an Aut   † NF φ , discussion of Example 4.3, (i)] of isomorphisms — which are compatible with  NF ‡ φ . There is an evident notion of composition of morphisms of D -NF-bridges.  , [relative to the given initial Θ ,or D - Θ -bridge -bridge (ii) We define a base- Θ-data] to be a poly-morphism Θ † φ  † † −→ D D J > † † † is a D -prime-strip ; } D , = { D D -prime-strips —where D capsule of is a J > j J ∈ j ∼ † → D , — such that there exist isomorphisms J indexed by a finite index set D > > ∼ † Θ Θ † D D φ → , conjugation by which maps  → φ [iso]morphism . We define a(n) J    - Θ -bridges of D ‡ † Θ Θ φ φ   † ‡ ‡ † ′ ( → D D ) ) D −→ ( −→ D J J > > to be a pair of poly-morphisms ∼ ∼ ‡ † † ‡ ′ D ; → D D D → J > > J ∼ ∼ ‡ † † ‡ ′ D is the D is a capsule-full poly-isomorphism ; full —where D → → D > J J > Θ ‡ Θ † φ . There is an evident , φ compatible — which are with poly-isomorphism   notion of composition of morphisms of D -Θ-bridges.

112 112 SHINICHI MOCHIZUKI ,or D - ΘNF -Hodge theater ,[relative ΘNF -Hodge theater base- (iii) We define a to the given initial Θ-data] to be a collection of data Θ † † NF φ φ   D -ΘNF † †  † † =( −→ ←− D ) D D HT J > NF † † Θ D -NF-bridge; φ φ is a -Θ-bridge — such that there exist is a D —where   isomorphisms ∼ ∼ ∼ †   † † ; → D ; D D → → D D D  > J > † NF Θ NF † Θ φ  - , φ conjugation by which maps D  → → φ φ [iso]morphism of .A(n)     is defined to be a pair of morphisms between the respective -Hodge theaters ΘNF D -NF- and D -Θ-bridges that are compatible with one another in the associated between the index sets of the respective sense that they induce the same bijection -prime-strips. There is an evident notion of composition of morphisms D capsules of of D -ΘNF-Hodge theaters. Proposition 4.7. (Transport of Label Classes of Cusps via Base- Bridges) Let Θ † NF † φ φ   D -ΘNF † † †  † =( HT ←− −→ D D D ) > J D - ΘNF -Hodge theater [relative to the given initial Θ -data]. Then: be a bad Θ † v (i) The structure at the various - Θ V D φ of the ∈ [i.e., in- -bridge  evaluation sections — cf. Example 4.4, (i), (ii); Definition 4.6, (ii)] volving determines a bijection ∼  † † F ( χ D : )= J π → J 0 l  F D -prime-strips of the capsule — i.e., determines ∈ for the constituent labels l † . D J [cf. Example 4.5] via the V ∈ at the various v (ii) For each j ∈ J , restriction  NF † Θ † , -torsors φ j determines an isomorphism of F φ indexed by portion of   l ∼ † LC †  † : LabCusp( ) φ → LabCusp( D D ) > j LC LC † † φ φ is obtained from [where, by abuse of notation, we write “ 1 ∈ J ” such that 1 j  † for the element of J that maps via 1 in F to the image of ] by composing with χ l  † the action by ( j ) ∈ F χ . l unique element (iii) There exists a  † † [ LabCusp( D ] ) ∈  ∼  † D of the ) → F ∈ LabCusp( J ,the j natural bijection such that for each > l LC † † LC † † † † ]) = φ φ .In j ( ) ([ ( j ) · [ χ  ])  →  χ ( second display of Proposition 4.2 maps 1 j  † -torsors [ particular, the element ] isomorphism of F  determines an l ∼ ∼   † † ) ) : LabCusp( → J ( ζ → F D  l

113 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 113 † χ of (i)] between [where the bijection in parentheses is the bijection “global ∼  † † cusps” [i.e., “ J  ] ”] and capsule indices j j ∈ ( ) → F · [ ]. Finally, χ [i.e., l  ,the F when considered up to composition with multiplication by an element of l  † † NF † NF bijection φ ζ within the F of the choice of is -orbit of independent φ    l  †  F relative to the natural poly-action of D [cf. Example 4.3, (iv); Fig. 4.4 on l below]. Assertion (i) follows immediately from the definitions [cf. Example 4.4, Proof. (i), (ii), (iv); Definition 4.6], together with the bijection of the second display of intrinsic Proposition 4.2. Assertions (ii) and (iii) follow immediately from the of the constructions of Example 4.5. © nature † of Proposition 4.7, ζ Remark 4.7.1. The significance of the natural bijection  (iii), lies in the following observation: Suppose that one wishes to work with the  † D in a fashion that is of the local data [i.e., “prime-strip independent global data † † D , D [cf. Remark 4.3.2, (b)]. Then data”] J > capsule index set J by the set of global label classes of by replacing the † †  †  ) via D , one obtains an object — i.e., LabCusp( D ζ ) LabCusp( cusps  global data that is immune to the — constructed via [i.e., “native to”] the ∼ ± un   of J “collapsing” — i.e., of → F even -orbits of V F at primes — l l V of the sort discussed in Remark 4.2.1! ∈ v   to act freely on] F -orbits F That is to say, this “collapsing” of [i.e., failure of l l ± un is a characteristically global consequence of the global prime decomposition V of trees discussed in Remark 4.3.1, (ii) [cf. the example discussed in Remark 4.2.1]. We refer to Remark 4.9.3, (ii), below for a discussion of a closely related phenomenon. Remark 4.7.2. labels [cf. the content of Proposition 4.7], the structure of a (i)Atthelevelof - -Hodge theater may be summarized via the diagram of Fig. 4.4 below — i.e., D ΘNF †   ]” corresponds to ;the D − 1)   † − 1 l D )” corresponds to ; the lower right- ... l expression “( 1 2 J   † hand “ F ’s” corresponds to D ;the“ ⇑ ” corresponds to the associated  -cycle of l  ’s” denote D ;the“ ⇒ ” corresponds to the associated D -NF-bridge ;the“ / - Θ -bridge D -prime-strips. † (ii) Note that the labels arising from D correspond, ultimately, to various > irreducible components in the special fiber of a certain tempered covering of a [a special fiber which consists of a [“geometric”!] Tate curve chain of copies of the projective line — cf. [EtTh], Corollary 2.9]. In particular, these labels are counting obtained by archimedean , additive fashion — the number —inanintuitive, of irreducible components between a given irreducible component and the “origin”. † D may That is to say, the portion of the diagram of Fig. 4.4 corresponding to > be described by the following terms: geometric , additive , archimedean , hence Frobenius-like [cf. Corol- lary 3.8].

114 114 SHINICHI MOCHIZUKI  By contrast, the various “ ’s” in the portion of the diagram of Fig. 4.4 correspond-  † primes of an [“arithmetic”!] number D arise, ultimately, from various ing to  × ,ina } F = 1 / {± F multiplicative group . These primes are permuted by the field l l nonarchimedean — fashion. Thus, the portion of the diagram of Fig. cyclic — i.e.,  † D may be described by the following terms: 4.4 corresponding to multiplicative , nonarchimedean , hence ́ etale-like [cf. arithmetic , the discussion of Remark 4.3.2]. † †  That is to say, the portions of the diagram of Fig. 4.4 corresponding to , D D > differ quite fundamentally in structure . In particular, it is not surprising that the only “common ground” of these two fundamentally structurally different portions  [i.e., the portion of the diagram of consists of an underlying set of cardinality l † D ]. Fig. 4.4 corresponding to J † — or, perhaps more appropriately, its inverse ζ (iii) The bijection  ∼ † 1  − † : ) J → LabCusp( ζ D ) (  arithmetic [i.e., if one thinks of the elements of — may be thought of as relating J the capsule index set collections of primes of a number field ]to geometry [i.e., as  † D ) as corresponding to the [geometric!] if one thinks of the elements of LabCusp( of the hyperbolic orbicurve]. From this point of view, cusps † − 1 ( “combinatorial Kodaira-Spencer may be thought of as a sort of ) ζ  [cf. the point of view of [HASurI], § 1.4]. morphism” † ζ . We refer to Remark 4.9.2, (iv), below, for another way to think about    <... Θ ⇑ φ  (  −→  1 ↗↘  / 2  NF φ  F  l . .  . /  ↑D = ↓ ⇒  1 − l 0 ) ( B  C )  K   D / l J ↖↙  /  ...  Fig. 4.4: The combinatorial structure of a D -ΘNF-Hodge theater The following result follows immediately from the definitions.

115 ̈ INTER-UNIVERSAL TEICHM 115 ULLER THEORY I Θ -Bridges, Proposition 4.8. (First Properties of Base-NF-Bridges, Base- -Hodge Theaters) ΘNF Θ - Relative to a fixed collection of and Base- initial data :  - (i) The forms an F between set of isomorphisms two D -NF-bridges l . torsor between (ii) The D - Θ -bridges (respectively, two set of isomorphisms two - -Hodge theaters )is of cardinality one . D ΘNF -NF-bridge and a D - Θ -bridge, the set of capsule-full poly- (iii) Given a D D isomorphisms between the respective capsules of -prime-strips which allow one to glue D -NF- and D - Θ -bridges together to form a D - ΘNF -Hodge theater the given  -torsor . F forms an l D (iv) Given a -NF-bridge, there exists a [relatively simple — cf. the discussion of Examples 4.4, (i), (ii), (iii); 4.5, (i), (ii)] functorial algorithm for construct-  ing, up to an F -indeterminacy -NF-bridge a D [cf. (i), (iii)], from the given l -NF-bridge. ΘNF -Hodge theater whose underlying D D D - -NF-bridge is the given Proposition 4.9. (Symmetries arising from Forgetful Functors) Relative to a fixed collection of initial Θ -data : (i) (Base-NF-Bridges) The operation of associating to a D - ΘNF -Hodge the- ater the underlying D - ΘNF -Hodge theater determines a natural -NF-bridge of the D functor category of category of D -NF-bridges D -ΘNF-Hodge theaters → and isomorphisms of and isomorphisms of D D -ΘNF-Hodge theaters -NF-bridges † NF φ  D -ΘNF † † †  ( HT D ←−  D → ) J  whose output data admits an F -symmetry which acts simply transitively on l † D ”] of D -prime-strips [i.e., “ the index set [i.e., “ J ”] of the underlying capsule of J this output data. The operation of associating to a - ΘNF - (Holomorphic Capsules) D (ii) Hodge theater the underlying capsule of D - ΘNF -Hodge theater D -prime-strips of the natural functor determines a  category of category of l -capsules -prime-strips D -ΘNF-Hodge theaters D of → and isomorphisms of and capsule-full poly-  D isomorphisms of l -ΘNF-Hodge theaters -capsules D -ΘNF † † HT  → D J

116 116 SHINICHI MOCHIZUKI   -symmetry [where we write S S whose output data admits an for the symmet- l l  letters] which acts transitively on the index set [i.e., “ J ”] of this ric group on l output data. Thus, this functor may be thought of as an operation that consists of ∼   → [i.e., forgetting the bijection J F of Proposition ∈ the labels forgetting F l l † D up to isomorphism, 4.7, (i)]. In particular, if one is only given this output data J   for the element ∈ F to which a possibilities l then there is a total of precisely l j ∈ J corresponds [cf. Proposition 4.7, (i)], prior to the application of given index this functor. (Mono-analytic Capsules) By composing the functor of (ii) with the (iii) operation discussed in Definition 4.1, (iv), one obtains a mono-analyticization natural functor  category of -capsules l category of  -prime-strips -ΘNF-Hodge theaters D of D → and capsule-full poly- and isomorphisms of  -ΘNF-Hodge theaters l D -capsules isomorphisms of D -ΘNF †  † D →  HT J whose output data satisfies the same symmetry properties with respect to labels as the output data of the functor of (ii). Assertions (i), (ii), (iii) follow immediately from the definitions [cf. also Proof. © Proposition 4.8, (i), in the case of assertion (i)]. Remark 4.9.1. (i) Ultimately, in the theory of the present series of papers [cf., especially, [IUTchII], 2], we shall be interested in § the ́ evaluating of [EtTh] — i.e., in the spirit of etale theta function the Hodge-Arakelov theory of [HASurI], [HASurII] — at the various † labels , in the fashion stipulated by the D discussed -prime-strips of D J in Proposition 4.7, (i). These values of the ́ arithmetic etale theta function will be used to construct various . We shall be interested in computing the arithmetic degrees —in line bundles the form of various — of these arithmetic line bundles. In order to “log-volumes” compute these global log-volumes, it is necessary to be able to compare the log- volumes that arise at D -prime-strips with different labels . It is for this reason that the output data of the functors of Proposition 4.9, (i), (ii), (iii) [cf. also non-labeled Proposition 4.11, (i), (ii), below], are of crucial importance in the theory of the present series of papers. That is to say, the non-labeled output data of the functors of Proposition 4.9, (i), (ii), (iii) [cf. also Proposition 4.11, (i), (ii), below] — which allow one to consider isomorphisms between the D -prime-strips that were originally

117 ̈ INTER-UNIVERSAL TEICHM 117 ULLER THEORY I different labels assigned of objects — make possible the comparison [e.g., log-volumes] constructed relative to different labels. “processions” , In Proposition 4.11, (i), (ii), below, we shall see that by considering minimizes the label indeter- one may perform such comparisons in a fashion that that arises. minacy  -symmetry that appears in Proposition 4.9, (i), is transitive , (ii) Since the F l it follows that one may use this action to perform comparisons as discussed in (i). This prompts the question:   What is the difference between this F -symmetry and the S -symmetry l l of the output data of the functors of Proposition 4.9, (ii), (iii)?  F Inaword,restrictingtothe -symmetry of Proposition 4.9, (i), amounts to the l  J on the index set F imposition of a “cyclic structure” [i.e., a structure of -torsor l  ]. Thus, relative to the issue of raised in (i), this J F on comparability -symmetry l between — i.e., involves isomorphisms between the non-labeled comparison allows J , without D -prime-strips corresponding to — distinct members of this index set J . This cyclic structure may be thought of as on disturbing the cyclic structure  † D that global object combinatorial manifestation a sort of of the link to the D -NF-bridge. On the other hand, appears in a compare D -prime-strips indexed by J “in the abso- in order to these to D -prime-strips that have nothing to do with J , it is necessary to lute” J “forget the cyclic structure on . ” This is precisely what is achieved by considering the functors of Proposition 4.9,  -symmetry”. (ii), (iii), i.e., by working with the “full S l Remark 4.9.2. (i) The various elements of the index set of the capsule of D -prime-strips of a  ”, D 2 ,...,l -NF-bridge are , synchronized in their correspondence with the labels “1 in the sense that this correspondence is completely determined up to composition  . In particular, this correspondence is always withtheactionofanelementof F l . bijective ,or cohesion One may regard this phenomenon of synchronization ,as important consequence of the fact that the number field in question is an D -NF-bridge via a single copy [i.e., as opposed to a represented in the  . whose index set is of cardinality 2] of D capsule ≥ † D is -prime-strip in the capsule each Indeed, consider a situation in which D J  ,towhichit , i.e., copy of D “independent globalization” equipped with its own NF ”, which [in order not to invalidate the comparability is related by a copy of “ φ j of distinct labels — cf. Remark 4.9.1, (i)] is regarded as being known only up to  F composition with the action of an element of . Then if one thinks of the [mani- l  -translates of festly mutually disjoint — cf. Definition 3.1, (f); Example 4.3, (i)] F l ⋂ ⋂ un Bor ± bad bad K K ( V ) V ( ) V ]asbeing labeled by the [whose union is equal to V  † elements of F ,then D -prime-strip in the capsule ” in Fig. D each — i.e., each “ • J l 4.5 below — is subject, as depicted in Fig. 4.5, to an independent indeterminacy

118 118 SHINICHI MOCHIZUKI  F to which it is associated. In particular, the set of all ∈ concerning the label l J possibilities for each association includes correspondences between the index set  † and the set of labels F which fail to be bijective .Moreover, D of the capsule J l  F although arises essentially as a subquotient of a Galois group of extensions of l  )], the fact that ( K on primes of V faithful poly-action of F number fields [cf. the l [cf. Example 4.3, (i)] implies that faithfully  it also acts on conjugates of the cusp  ( K ) up to F “working with elements of V -indeterminacy” may only be done at the l  up to -indeterminacy” .Thatis F  expense of “working with conjugates of the cusp l with the construction to say, “working with nonsynchronized labels” is inconsistent † ζ in Proposition 4.7, (iii). crucial bijection of the   ? 1? 2? 3? ··· l •  →   1? 2? 3? ··· l → • ? . . . . . .  ? → 1? 2? 3? ··· l •  Fig. 4.5: Nonsynchronized labels (ii) In the context of the discussion of (i), we observe that the “single copy” of  may also be thought of as a “single connected component” , hence — from D —asa Galois categories . the point of view of “single basepoint” (iii) In the context of the discussion of (i), it is interesting to note that since   on F is transitive , one obtains the same “set of all F the natural action of l l possibilities for each association”, regardless of whether one considers independent   at each index of J or independent S -indeterminacies -indeterminacies at each F l l J [cf. the discussion of Remark 4.9.1, (ii)]. index of synchronized indeterminacy [cf. (i)] exhibited by a D -NF-bridge (iv) The † ζ of Proposition 4.7, (iii) — crucial bijection — i.e., at a more concrete level, the  may be thought of as a sort of of the notion of a “holo- combinatorial model . By contrast, the dis- morphic structure” nonsynchronized indeterminacies cussed in (i) may be thought of as a sort of combinatorial model of the notion of “real analytic structure” a . Moreover, we observe that the theme of the above discussion — in which one considers “how a given combinatorial holomorphic structure is ‘embedded’ within its underlying combinatorial real analytic structure” — is very much in line with the spirit of classical complex Teichm ̈ uller theory . (v) From the point of view discussed in (iv), the of the “multi- main results plicative combinatorial Teichm ̈ uller theory” developed in the present § 4may be summarized as follows: (a) globalizability of labels , in a fashion that is independent of local structures [cf. Remark 4.3.2, (b); Proposition 4.7, (iii)]; (b) comparability of distinct labels [cf. Proposition 4.9; Remark 4.9.1, (i)]; (c) absolute comparability [cf. Proposition 4.9, (ii), (iii); Remark 4.9.1, (ii)];

119 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 119 minimization of label indeterminacy — without sacrificing the symmetry (d) necessary to perform comparisons! — via processions [cf. Proposition 4.11, (i), (ii), below]. Remark 4.9.3. (i) Ultimately, in the theory of the present series of papers [cf. [IUTchIII]], we would like to apply the of [AbsTopIII] to the various mono-anabelian theory ,Π for local and global arithmetic fundamental groups [i.e., isomorphs of Π v C K non ∈ V D -ΘNF-Hodge theater [cf. the discussion of Remark ] that appear in a v 4.3.2]. To do this, it is of essential importance to have available not only the absolute Galois groups of the various local and global base fields involved, but ,Π geometric fundamental groups also the that lie inside the isomorphs of Π v C K involved. Indeed, in the theory of [AbsTopIII], it is precisely the outer Galois action of the absolute Galois group of the base field on the geometric fundamental group that allows one to reconstruct the ring structures group-theoretically in a fashion localization/globalization operations as shown in Fig. 4.3. that is compatible with Here, we pause to recall that in [AbsTopIII], Remark 5.10.3, (i), one may find a discussion of the analogy between this phenomenon of “entrusting of arithmetic moduli” [to the outer Galois action on the geometric fundamental group] and the Kodaira-Spencer isomorphism of an indigenous bundle — an analogy that is reminiscent of the discussion of Remark 4.7.2, (iii). (ii) Next, let us observe that the state of affairs discussed in (i) has important [i.e., ” circumstances that necessitate the use of “ X implications concerning the − → v as opposed to “ C ”] in the definition of “ D ” in Examples 3.3, 3.4 [cf. Remark v v localization/globalization operations as shown in Fig. 4.3 give rise, 4.2.1]. Indeed, when applied to the various geometric fundamental groups involved, to various between local and global sets of label classes of cusps . Now suppose that bijections ” in Examples 3.3, 3.4. ” instead of “ X D in the definition of “ ” one uses “ C v v → − v v Then the existence of V of the sort discussed in Remark 4.2.1, together with ∈ the condition of compatiblity with localization/globalization operations as shown in Fig. 4.3 — where we take, for instance, def v =( of Remark 4.2.1) ( of Fig. 4.3) v def ′ v ( =( of Fig. 4.3) of Remark 4.2.1) w — imply that, at a combinatorial level, one is led, in effect, to a situation of the sort discussed in Remark 4.9.2, (i), i.e., a situation involving nonsynchronized labels [cf. Fig. 4.5], which, as discussed in Remark 4.9.2, (i), is incompatible with † ζ of Proposition 4.7, (iii), an object which the construction of the crucial bijection will play an important role in the theory of the present series of papers. Definition 4.10. Let be a category , n a positive integer. Then we shall refer C to as a procession of length n ,or n -procession ,of C any diagram of the form P ↪ ↪ → P P → → ... ↪ 1 2 n

120 120 SHINICHI MOCHIZUKI [for j =1 ,...,n ]isa j -capsule [cf. § 0] of objects of C ; each —whereeach P j ,...,n P [for j =1 → − 1] denotes the collection of all capsule-full ↪ arrow P j +1 j morphism to P to .A from an n -procession of C [cf. § poly-morphisms P 0] from +1 j j -procession of C an m ( P ↪ → ) → ( Q ...↪ P → ...↪ → Q ) ↪ → 1 n m 1 ι : { 1 ,...,n } ↪ →{ 1 ,...,m } [so n ≤ m ], consists of an order-preserving injection ↪ → Q . ,...,n =1 j for each P together with a capsule-full poly-morphism j ) ι j (         → ↪ / / / / ↪ → ... ↪ → ( / → ... / / ) ↪ /  -procession of D -prime-strips Fig. 4.6: An l Proposition 4.11. (Processions of Base-Prime-Strips) Relative to a fixed initial Θ -data : collection of † † Θ † -bridge (Holomorphic Processions) Given a - Θ D (i) : D → D φ J >  † -prime-strips [cf. Definition 4.6, (ii)], with underlying capsule of D D , denote J †  the [cf. Fig. 4.6, where each ) -prime-strips l D -procession of D Prc( by J  ” denotes a D -prime-strip] determined by considering the [“sub”]capsules of “ / def def     † ... S D S ⊆ = { 1 , 2 ,...,j }⊆ corresponding to the subsets ⊆ S ... F = ⊆  J 1 j l l [where, by abuse of notation, we use the notation for positive integers to denote the ∼   † images of these positive integers in F : J χ → F ], relative to the bijection of l l † † Θ  → Prc( φ natural determines a ) D Proposition 4.7, (i). Then the assignment J  functor category of processions category of of D -prime-strips D -Θ-bridges → and morphisms of and isomorphisms of D -Θ-bridges processions † † Θ  Prc( φ D ) → J   n 1 ,...,l whose output data satisfies the following property: for each ∈{ } ,there  possibilities for the element ∈ F are precisely n to which a given index of the l n -capsule that appears in the procession constituted by this output index set of the data corresponds, prior to the application of this functor. That is to say, by tak-  , of cardinalities of “sets of possibilies”, one F ing the product, over elements of l concludes that by considering processions — i.e., the functor discussed above, possi- -ΘNF D † Θ † HT  → φ that associates to bly pre-composed with the functor  D - ΘNF -Hodge theater its associated D - Θ -bridge — the indeterminacy a   ) ( l l consisting of ( ) possibilities that arises in Proposition 4.9, (ii), is  possibilities ! . l consisting of a total of indeterminacy to an reduced (ii) (Mono-analytic Processions) By composing the functor of (i) with the mono-analyticization operation discussed in Definition 4.1, (iv), one obtains a

121 ̈ INTER-UNIVERSAL TEICHM 121 ULLER THEORY I natural functor category of category of processions  -Θ-bridges of D -prime-strips D → and isomorphisms of and morphisms of -Θ-bridges D processions Θ †  † Prc( ) D φ  →  J whose output data satisfies the same indeterminacy properties with respect to labels as the output data of the functor of (i). Assertions (i), (ii) follow immediately from the definitions. Proof. © The following result is an immediate consequence of our discussion. ́ ΘNF -Hodge Theaters) Corollary 4.12. ( Relative Etale-pictures of Base- Θ -data : to a fixed collection of initial (i) Consider the [composite] functor D -ΘNF † †  † D  →  D HT → > > — from the category of ΘNF -Hodge theaters and isomorphisms of D - ΘNF - D -Hodge  -prime-strips and isomor- D theaters [cf. Definition 4.6, (iii)] to the category of  ΘNF -prime-strips — obtained by assigning to the D - -Hodge theater D phisms of -ΘNF D †  † HT D D mono-analyticization the of the [cf. Definition 4.1, (iv)] - > † D [cf. that appears as the codomain of the underlying D - Θ -bridge prime-strip > -ΘNF D -ΘNF D -ΘNF D † ‡ † Definition 4.6, (ii)] of HT HT HT , are D - ΘNF -Hodge .If , then we define the base- -, or D - ΘNF -, link theaters ΘNF D -ΘNF D -ΘNF D ‡ † −→ HT HT -ΘNF D -ΘNF D † ‡ HT from to be the full poly-isomorphism HT to ∼  ‡ †  D D → > >  between the D -prime-strips obtained by applying the functor discussed above to -ΘNF D D -ΘNF ‡ † HT , . HT (ii) If D D D D D -ΘNF D -ΘNF -ΘNF D n 1) − n ( +1) ( n ... −→ HT HT HT −→ −→ ... −→ [where n ∈ Z ]isan infinite chain of D - ΘNF -linked D - ΘNF -Hodge theaters [cf. the situation discussed in Corollary 3.8], then we obtain a resulting chain of full poly-isomorphisms ∼ ∼ ∼ n  ( n +1)  → → D D → ... ... > >

122 122 SHINICHI MOCHIZUKI  D [cf. the situation discussed in Remark 3.8.1, (ii)] between the -prime-strips ob- tained by applying the functor of (i). That is to say, the output data of the functor constant invariant of (i) forms a [cf. the discussion of Remark 3.8.1, (ii)] — i.e., a mono-analytic core [cf. the situation discussed in Remark 3.9.1] — of the above infinite chain. - ΘNF -Hodge theaters of the chain of (ii) as a (iii) If we regard each of the D emanating from the mono-analytic core discussed in (ii), then we obtain a spoke ́ etale-picture of D — i.e., an ΘNF -Hodge theaters — as in Fig. 4.7 diagram -  ” denotes below [cf. the situation discussed in Corollary 3.9, (i)]. In Fig. 4.7, “ >    / ” denotes the “holomorphic” proces- ↪ → ↪ → ... / the mono-analytic core; “ / sions of Proposition 4.11, (i), together with the remaining [“holomorphic”] data of the corresponding - ΘNF -Hodge theater. Finally, [cf. the situation discussed in D Corollary 3.9, (i)] this diagram satisfies the important property of admitting arbi- trary permutation symmetries among the spokes [i.e., among the labels n ∈ Z of the D - ΘNF -Hodge theaters].    ... / / ↪ → → ↪ / ... ... | — —        ↪ ... > ↪ / / / → → → / ↪ / / ↪ → ... | ... ...    ↪ / / / ↪ → → ... ́ Fig. 4.7: Etale-picture of D -ΘNF-Hodge theaters

123 ̈ INTER-UNIVERSAL TEICHM 123 ULLER THEORY I Section 5: ΘNF -Hodge Theaters In the present 5, we continue our discussion of various “enhancements” to the § -Hodge theater Θ-Hodge theaters of § 3. Namely, we define the notion of a ΘNF [cf. Definition 5.5, (iii)] and observe that these ΘNF-Hodge theaters satisfy the [cf. Corollary 5.6; Remark 5.6.1] as the base-ΘNF- same “functorial dynamics” Hodge theaters discussed in § 4. Let Θ † NF † φ φ   D -ΘNF † † † †  D D −→ ←− D =( ) HT > J - ΘNF -Hodge theater [cf. Definition 4.6], relative to a fixed collection of initial be a D bad V ,l,C , V , ) as in Definition 3.1. F/F, X , Θ ( -data F K mod Example 5.1. Global Frobenioids. anabelian result of [AbsTopIII], Theorem 1.9, via the (i) By applying the  † group- D ), we may construct ( “ Θ π discussed in Remark 3.1.2, to -approach” 1 × †  π theoretically from ” — which we shall denote )anisomorphof“ ( D F 1 †   M ) D ( †  natural π — equipped with its ( D -action . Here, we recall that this construction ) 1 ⋃  def †   †  ) D = M ( ( . D } ) M { 0 on includes a reconstruction of the field structure F C -core of the Next, let us recall [cf. Remark 3.1.7, (i)] the C unique model F F mod †  . Observe that one may construct group-theoretically from π ), in a ( F D over 1 mod ” [cf. the algorithms of profinite group corresponding to “ C functorial fashion, a F mod [AbsTopII], Corollary 3.3, (i), which are applicable in light of [AbsTopI], Example  †  † 0 open subgroup ;write D D ( for B ( − ) )asan of this 4.8], which contains π 1 profinite group, so we obtain a natural morphism  †  † D D → — i.e., a “category-theoretic version” of the natural morphism of hyperbolic orbi-  † ) D ( natural extension π C curves → C — together with a of the action of 1 F K mod   † †   † on M ( )to D D ). In particular, by taking π π ( , we obtain D ( ) -invariants 1 1 a submonoid/subfield     †   † †   † ⊆ ( , M ⊆ D ( ) D D ) ) M ( ( M D ) M mod mod × × F corresponding to F , F ⊆ F . In a similar vein, by applying [AbsTopIII], ⊆ mod mod Belyi cuspidalizations of [Ab- Theorem 1.9 — cf., especially, the construction of the × ” of [AbsTopIII], Theorem } ∪{ 0 field “ K sTopIII], Theorem 1.9, (a), and of the Z NF †  π group-theoretically 1.9, (d), (e) — we conclude that we may construct from D ), ( 1 in a functorial fashion, an isomorph  rat †  † π D )(  π ( ( )) D 1 1

124 124 SHINICHI MOCHIZUKI of the absolute Galois group of the function field of [i.e., equipped with its C F mod  † ( D ) and well-defined up to inner automorphisms deter- natural surjection to π 1 mined by elements of the kernel of this natural surjection], as well as isomorphs associated to κ -, and -coric rational functions κ × pseudo-monoids of of the κ -, ∞ ∞ [cf. the discussion of Remark 3.1.7, (i), (ii); [AbsTopII], Corollary 3.3, (iii), C F mod which is applicable in light of [AbsTopI], Example 4.8] — which we shall denote   †  †  †   ) ( ) D , ) , M M D D ( ( M κ κ × κ ∞ ∞  rat †  †  D .Thus, M ( ( ) D )maybeiden- -actions π — equipped with their natural κ 1 †     †  rat † ( D )may ( ) D -invariants ), and M of ( M D tified with the subset of π κ 1 ∞ be identified with a certain subset [i.e., indeed, a certain “sub-pseudo-monoid”!]  †  ). Next, let us observe that it also follows from the group-theoretic D ( of M κ × ∞ †  D ), ( constructions recalled above that one may reconstruct the quotients of π 1 †  D ( ) that correspond, respectively, to the absolute Galois groups of K , F . π mod 1 rat †  ( of the )bythe intersection of the kernel D π Thus, by forming the quotient of 1 †   rat † †  rat  ( ( )on D M ) with the inverse image in π D D ( )ofthekernel action of π κ 1 1 ∞  rat † D ( ) that corresponds to] maximal solvable quotient π of the of [the quotient of 1 ,weobtaina construction for a group-theoretic F the absolute Galois group of mod quotient rat   κ -sol † † π ( D ) ( ) D  π 1 1 -sol /κ rat †  π kernel we denote by —whose D quotient ( ) — that corresponds to the 1 L of Remark 3.1.7, (iv), as well as /L ” )  Gal( pseudo- ) ( κ -sol) /L L Gal( “ C C C C  † -sol κ ) D ( -actions equipped with natural π monoids 1    † † †    ) M D ( ( D D , , M ) ( ) M κ -sol sol κ ∞ -sol rat /κ   †  †   † M —where - D π ( M ), ) denote the respective subsets of ( D D ( ) 1 κ sol -sol  †   †  M invariants of ( M ), ( D D characterization ). Moreover, by applying the × κ ∞ K/F ⊆ given in Remark 3.1.7, (iii), we ” ) ) Gal( ( of the subgroup “ μ Gal( K/F mod l group-theoretic construction for subgroups obtain a SL †  SL †  †  D ( ) D ) ( ⊆ Aut( ⊆ D Aut ) Aut  SL SL that correspond to the subgroups “ Aut C of Ex- ) ⊆ Aut ( ( C ” ) ⊆ Aut( C )  K K K ample 4.3, (i), hence induce natural isomorphisms ∼ ∼   †  SL † †  †  SL D ( ) ) / Aut( Aut D → ) / Aut → ( F D ( ) D Aut   l — i.e., which, in the spirit of Example 4.3, (iv), may be thought of as a poly-   †  F action of on . Finally, we observe that although this poly-action of F on D l l rat  † ( )is only well-defined up to conjugation by elements of the subgroup D π 1 def †  rat rat  †  † = π D D ( ) D ) ) × ( ( π π †  1 ( π ) D 1 1 1 rat †  ( linear disjointness D property discussed in ), it follows formally from the of π 1  F Remark 3.1.7, (iii), that, by regarding this poly-action of ac- as arising from the l rat /κ -sol †   SL † tion of elements of Aut D ), one may conclude that, if we write ) ( π D ( 1 ⋂ def rat /κ -sol † rat  †  D ), then ( π = ) ( π D 1 1

125 ̈ INTER-UNIVERSAL TEICHM 125 ULLER THEORY I   † rat well-defined π ( on D of ) is, in fact, F the resulting poly-action 1 l rat /κ -sol  † up to π D ) -conjugacy indeterminacies , hence, in particular, ( 1 codomain ,and arrow that that the induced poly-action on [the domain , -sol -outer representation” κ “ constitute] the -sol /κ rat † †  -sol κ  κ -sol )) ( → ) π ( D D ( Out π 1 1 — i.e., which may be associated to and is, in fact, equivalent to the exact rat /κ -sol -sol †  rat †  †  κ π sequence 1 → ( ) D D 1, regarded → ) → π ) ( ( D → π 1 1 1 -sol rat /κ †  up to π D -conjugacy indeterminacies [cf. the discussion of Re- ( ) 1 mark 3.1.7, (iv)] — is, in fact, well-defined without any conjugacy trivial action . indeterminacies , and, moreover, equal to the We shall refer to this phenomenon [cf. also Remark 5.1.5 below] as the phenomenon κ -conjugate synchronization . -sol of (ii) Next, let us recall [cf. Definition 4.1, (v)] that the field structure on    † † ) [i.e., “ F ”] allows one to reconstruct group-theoretically from ( ) M D D π ( 1   † †  set of valuations the V D F )”] on M ) [i.e., “ ( D ( V ) equipped with its natural ( †  †  on )-action, hence also the monoid [i.e., in the sense of [FrdI], Definition D D ( π 1 1.1, (ii)]   † Φ ( )( − ) D   †  † monoid Φ )the ( D D )( A ) of “stack- A Ob( that associates to an object ∈ arithmetic divisors on the corresponding subfield [cf. Remark 3.1.5] theoretic”   † †   A ⊆ )” in [FrdI], Example 6.3; ) ( D D ( ) [i.e., the monoid denoted “Φ( − M M cf. also Remark 3.1.5 of the present paper], together with the natural morphism gp   † †   A ( → ( D D ) )( A ) [cf. the discussion of [FrdI], Example Φ of monoids M 6.3; Remark 3.1.5 of the present paper]. As discussed in [FrdI], Example 6.3 [cf. also Remark 3.1.5 of the present paper], this data determines, by applying [FrdI], model Frobenioid Theorem 5.2, (ii), a   † D ) ( F †  over the base category D .  † †   †  F equivalent to category D be any ). Thus, F F ( is equipped (iii) Let with a natural Frobenioid structure [cf. [FrdI], Corollary 4.11; [FrdI], Theorem 6.4,  † F base category of ) for the (i); Remark 3.1.5 of the present paper]; write Base( this Frobenioid. Suppose further that we have been given a morphism   † † Base( F D ) →  †  † abstractly equivalent § [cf. which is 0] to the natural morphism → D D [cf. (i)]. In the following discussion, we shall use the resulting [ uniquely determined , in light , together with [AbsTopIII], Theorem 1.9!] isomorphism of the of C -coricity F F ∼  † †  †  †  Base( to identify Base( F F → )with D D ) . Let us denote by def †  †   † = | F ) F ( → F  † D

126 126 SHINICHI MOCHIZUKI  †  †  †  † restriction to D F → D D the and by of via the natural morphism def  † †   † = ⊆ | F F ) ( F terminal objects mod †  †  to the full subcategory of determined by the terminal D F of the restriction †  D ” ”] of . Thus, when the data denoted here by the label “ † C objects [i.e., “ F mod arises [in the evident sense] from data as discussed in Definition 3.1, the Frobenioid  † F may be thought of as the Frobenioid of arithmetic line bundles on the stack mod ” of Remark 3.1.5. S “ mod (iv) We continue to use the notation of (iii). We shall denote by a superscript [which are birationalizations — cf. [FrdI], Corollary “birat” the category-theoretic 4.10; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper] of the Frobe-  †  † †  F F , F , ; we shall also use this superscript to denote the images nioids mod of objects and morphisms of these Frobenioids in their birationalizations. Thus, if ×  birat † F O A ), then ) may be naturally identified with the multiplicative ( ∈ Ob( A ] cor- F group of nonzero elements of the number field [i.e., finite extension of mod . In particular, by allowing Frobenius-trivial to vary among the responding to A A category-theoretic — cf. [FrdI], Definition 1.2, (iv); [FrdI], [a notion which is objects Corollary 4.11, (iv); [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper] † †   Galois objects of F D that lie over ,weobtaina pair [i.e., consisting of a of topological group acting continuously on a discrete abelian group] † ×   ̃  O ) D ( π 1 — which we consider up to the action by the “inner automorphisms of the pair”  † of the ( D divisor monoid ). Write Φ for the arising from conjugation by π †  1 F †  [which is category-theoretic — cf. [FrdI], Corollary 4.11, (iii); F Frobenioid [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper]. Thus, for each 0], § )” as in [FrdI], − )) [where we use the notation “Prime( A ( Prime(Φ ∈ p  † F × gp birat ) → Φ A ( [cf. [FrdI], Proposition 4.4, ( A ) O the natural homomorphism  † F (iii)] determines — i.e., by taking the inverse image via this homomorphism of submonoid ( A ) constituted by p —a } of] the subset of Φ [the union with { 0 †  F  birat × ⊆O ( A ). That is to say, in more intuitive terms, this submonoid is the O p birat × ( A ) with respect to the valuation de- submonoid of integral elements of O ×  p A .Write O of the number field corresponding to termined by for the ⊆O p p invertible elements . Thus, by allowing A to vary among the Frobenius- submonoid of †  †  that lie over Galois objects of and considering the way D F trivial objects of birat × ) ( ( A )on O the various sub- permutes A in which the natural action of Aut  † F  †  ∈ p A ∈ Prime(Φ F ) Ob( ( A , it follows that for each )), where monoids O  † 0 0 0 F p †  D ,weobtaina closed subgroup [well-defined up to terminal object lies over a of conjugation]  † ⊆ ( D ) π Π 1 p 0  by considering the elements of Aut fix ( A )that ,forsome the submonoid O †  F p . That is to say, in more intuitive terms, the subgroup p p system of ’s lying over 0 .Inparticular, V ∈ is simply the decomposition group associated to some v Π mod p 0 -cohomological dimension p if and only if the is nonarchimedean p it follows that 0 is equal to 2 + 1 = 3 for infinitely many prime numbers p [cf., e.g., [NSW], of Π p 0 Theorem 7.1.8, (i)].

127 ̈ INTER-UNIVERSAL TEICHM 127 ULLER THEORY I (v) We continue to use the notation of (iv). Let us write   † † † †  † κ -sol   π M ,π M )   D ) ( , D M ( 1 1 mod sol  † ×  ̃ ( ) O π D discussed in (iv) and its respective subsets [i.e.,  for the pair 1 rat /κ -sol   † rat  †  † † , ( D ]of )-, π M )-invariants. We shall refer to a pair ( M D π 1 1 sol mod rat †  ( D )] [i.e., consisting of a pseudo-monoid equipped with a continuous action by π 1    †  † rat rat † † π ) (respectively, π  ( M M D D )  ) ( 1 1 κ κ × ∞ ∞  † as an (respectively, on κ × -coric ) structure if it is isomorphic [i.e., κ -coric F ∞ ∞ pseudo-monoid equipped with a continuous action by as a pair consisting of a  rat † D )] to the pair ( π 1   rat  †  † † rat †   π ) ) M (respectively, )) D ( D D ( ) M ( D ( π 1 1 κ × κ ∞ ∞ rat †  π of (i). Thus, the D - ) -action that appears in an × κ -coric (respectively, ( κ ∞ ∞ 1 coric) structure necessarily ) through the natu- factors does not factor (respectively, †  rat κ -sol  † ( π  D ) of (i). Suppose that we have been given an ( ) D π ral surjection 1 1 †  (respectively, κ )” is a [commutative] × -coric ) structure on κ F -coric .If“( − ∞ ∞ , then let us write monoid def / = Hom( − )) )) − Q (( Z , ( μ ̂ Z [cf. [AbsTopIII], Definition 3.1, (v); [AbsTopIII], Definition 5.1, (v)]; note that this notational convention also makes sense if “( − cyclotomic pseudo-monoid [cf. )” is a Θ †  ( π )” of [AbsTopIII], ( D μ (Π )) for the cyclotome “ § μ 0]. Also, let us write 1 − ( ) ̂ Z ̂ Z Θ [cf. Remark “via the -approach” Theorem 1.9, which we think of as being applied  †   †  †  π 3.1.2] to )) D ( ( ). Then let us D observe ) (respectively, M that D M ( 1 κ × κ ∞ ∞ constructed [cf. [AbsTopIII], Theorem 1.9, (d)] as a subset of is, in effect, 1 Θ  † ( μ D H ))) ( π H, ( lim 1 − → ̂ Z H † rat † κ -sol   π ranges over the open subgroups of H —where π ( )). ( D D ) (respectively, 1 1  κ -sol † ) D ( On the other hand, consideration of π Kummer classes [i.e., of the action of 1 rat †  -th roots of elements, for positive integers ( ] yields D N )) on N (respectively, π 1  †  † of a natural injection M (respectively, M )into κ × κ ∞ ∞  1  1 † † lim )) H, ( ( M H H, ( μ (respectively, lim M ))) ( H μ κ κ × ∞ ∞ − → − → ̂ ̂ Z Z H H † rat  κ -sol  † π ranges over the open subgroups of —where H ) (respectively, π )), D ( ( D 1 1 injectivity follows immediately from the corre- and we observe that the asserted     † † D ) (respectively, )). In par- M D ( ( M sponding injectivity in the case of κ κ × ∞ ∞ ticular, it follows immediately, by considering divisors of zeroes and poles [cf. the definition of a “ -coric function ” given in Remark 3.1.7, (i)] associated to Kum- κ mer classes of rational functions as in [AbsTopIII], Proposition 1.6, (iii), from the ̂ Q ↪ → Q Z ⊗ elementary observation that, relative to the natural inclusion , ⋂ × ̂ Z 1 { } = Q > 0

128 128 SHINICHI MOCHIZUKI unique isomorphism of cyclotomes that there exists a ∼ ∼  †  † †  Θ Θ  † ( ) ( μ M → ( M ( (respectively, μ )) π μ )) π → ( D D ( )) μ 1 1 κ κ × ∞ ∞ ̂ ̂ Z Z ̂ ̂ Z Z such that the resulting isomorphism between direct limits of cohomology modules isomorphism as considered above induces an ∼ ∼    †   †  † † M M (respectively, ) ( → → ) ( M D D ) M κ κ × κ κ × ∞ ∞ ∞ ∞ †  rat π equipped with continuous actions by pseudo-monoids [i.e., of ( D )]. In a 1 similar vein, it follows immediately from the theory summarized in [AbsTopIII], Theorem 1.9, (d), that there exists a unique isomorphism of cyclotomes ∼ Θ  †  † ( D π )) ) → μ M ( ( μ 1 ̂ Z ̂ Z such that the resulting isomorphism between direct limits of cohomology modules induces isomorphisms ∼ ∼ ∼     †    † † † † †   D M ( → , M M ) , M M → ( ( D → ) D ) M sol mod mod sol †  π ( D )] in a fashion that is [i.e., of monoids equipped with continuous actions by 1  ” [cf. the discussion of (iv)], relative to with the O compatible “ integral submonoids p the ring structure domains of constructed in [AbsTopIII], Theorem 1.9, (e), on the these isomorphisms. In particular, it follows immediately from the above discussion that †  always admits an F ,which κ -coric (respectively, structure κ × -coric ) ∞ ∞ uniquely determined isomorphism [i.e., of is, moreover, unique up to a rat †  D )]. ( equipped with continuous actions by pseudo-monoids π 1 , this uniquely de- Thus, in the following, we shall regard, without further notice †  κ × -coric ) structure on -coric F (respectively, as a collection of κ termined ∞ ∞ †  F . Here, we observe that the various iso- naturally associated to data that is †  M , morphisms of the last few displays allow one to regard the pseudo-monoids κ ∞  †  †  †  † F as being M [cf. the definition of via M related to the Frobenioid M × κ ∞ at the beginning of the present (v)] and the morphisms ∼   † †  †  × × † †  , ( → M M M ) ) M ↪ → ( M → ↪ κ κ κ × κ × ∞ ∞ ∞ ∞ induced by the various isomorphisms of the last few displays, together with the corresponding inclusions/equalities   † †   M ( ⊆ ) ( , ) M D D κ × κ ∞ ∞   †  ×   † †  × M ( ) D D M ( ( )) D ( )) ⊆ = M ( κ × κ ∞ ∞ — where we use the superscript “ × ” to denote the subset of invertible elements of a pseudo-monoid [cf. the discussion of the initial portion of (i)]. Also, we shall write   †   † † † M M , M M ⊆ ⊆ κ κ κ × -sol κ ∞ ∞

129 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 129 -sol /κ rat †  rat †  -invariants π ) for the respective “sub-pseudo-monoids” of ( ( D D ) -, . π 1 1 In this context, we observe further that it follows immediately from the discus- sion of Remark 3.1.7, (i), (ii), (iii) [cf. also [AbsTopII], Corollary 3.3, (iii), which is applicable in light of [AbsTopI], Example 4.8], and the theory summarized in [AbsTopIII], Theorem 1.9 [cf., especially, [AbsTopIII], Theorem 1.9, (a), (d), (e)], that  † rat †   † π D × )  κ M determines -coric structure F ( on any ∞ 1 × κ ∞ -coric structure κ an associated ∞  † †   rat † π  D M ) ⊆ ( M κ κ × 1 ∞ ∞ by considering the subset of elements for which the restriction of the as- sociated Kummer class [as in the above discussion] to some [or, equiva-  rat † ( D ) that corresponds to an open subgroup π lently, every] subgroup of 1 of the of some strictly critical point of C decomposition group F mod [i.e., corresponds to a root of unity], is a torsion element and, moreover, that the operation of restricting Kummer classes [as in the above discus-  †  † † -sol  κ sion] arising from D π M M ( ⊆ to subgroups of )thatcor- κ κ 1 ∞ -, F -valued F respond to decomposition groups of non-critical sol mod the reconstructing yields a functorial algorithm for C of points F mod    † † -sol † κ , together with the -action ( M ) D , field M π monoids with 1 mod sol structure — and hence, in particular, the topologies determined by the   † † -coric κ , ,fromthe M M } with { 0 — on the union of valuations ∞ sol mod †  F . associated to structure A similar statement to the statement of the last display holds, if one makes the following substitutions: κ -sol  † † rat  ( )” “ π D )”; (  D π “ 1 1    † † † “ F M “ -” M  ”. , F M -, -”; ”  “ F “ mod sol sol mod In particular, we obtain a purely construction, from the category category-theoretic †  natural bijection F ,ofthe ∼  † V ) F → Prime( mod mod  † F ) for the set of primes [cf. [FrdI], § — where we write Prime( 0] of the divisor mod  †  †  † monoid of as the set of π D ( ; we think of D ). Now, ) -orbits of V ( F V 1 mod mod in the notation of the discussion of (iv), suppose that is p nonarchimedean [i.e., lies ]. Thus, p valuation , hence, in particular, a determines a p over a nonarchimedean 0 ×  × birat 0 }∪O { ring on the topology O A ). Write , , O ( completions for the respective p p ̂ ̂ ×   O may be identified O .Then , with respect to this topology, of the monoids O p p p ̂ with the multiplicative monoid of nonzero integral elements of the completion of the number field corresponding to A at the prime of this number field determined by p . Thus, again by allowing A to vary and considering the resulting system of ind-  nonarchimedean ,ofthe p ”, we obtain a construction, for topological monoids O “ 0 p ̂ pair [i.e., consisting of a topological group acting continuously on an ind-topological monoid]

130 130 SHINICHI MOCHIZUKI  ̃  O Π p 0 ̂ p 0  † in ( commensurably terminal D is ) — cf., e.g., [AbsAnab], π — which [since Π 1 p 0 Theorem 1.1.1, (i)] we consider up to the action by the “inner automorphisms of . In the language of [AbsTopIII], § 3, this the pair” arising from conjugation by Π p 0 “MLF-Galois TM -pair of strictly Belyi type” [cf. [AbsTopIII], Definition pair is an 3.1, (ii); [AbsTopIII], Remark 3.1.3]. (vi) Before proceeding, we observe that the discussion of (iv), (v) concerning  † †   † †  F F D D . We leave the routine details to , , may also be carried out for the reader. † J -prime-strips D index set of the capsule of (vii) Next, let us consider the . D J ∼ def natural bijection .Thus,wehavea → V = { v , i.e., } V V ,write J ∈ For j v ∈ V j j j  → v . These bijections determine a “diagonal subset” v given by sending j ∏ def V V V ⊆ = J 〉 j J 〈 j ∈ J ∼ V natural bijec- — which also admits a .Thus,weobtain natural bijection → V 〉 〈 J tions ∼ ∼ ∼  † V V → → Prime( ) F → V mod 〉 J j 〈 mod j ∈ J .Write for def ∼    † † † { } F ) F , F V Prime( → = 〉 〈 J mod mod J 〈 〉 def ∼    † † † Prime( V F } → { = F F ) , j j mod mod   † † F as a copy of F “situated on” the . That is to say, we think of j J ∈ for j mod   † † † F D F ; we think of as a copy of j of the capsule constituent labeled J mod 〈 J 〉 † all the constituents of the capsule D .Thus, “situated in a diagonal fashion on” J we have a natural embedding of categories ∏ def    † † † F F → ↪ F = j J 〉 J 〈 ∈ j J  † underlying category — where, by abuse of notation, we write of [i.e., for the F 〉 J 〈  † not regard the F . Here, we remark that we do the first member of the pair] 〉 J 〈  † category F as being equipped with a Frobenioid structure .Write J ∏ def  R R  R  R  † † † † F F F F ; ; = j j J J 〈 〉 J ∈ j for the respective realifications [or product of the underlying categories of the re- alifications] of the corresponding Frobenioids whose notation does not contain a superscript “ R ”. [Here, we recall that the theory of realifications of Frobenioids is discussed in [FrdI], Proposition 5.3.]

131 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 131 · · v n ◦ ... ... ◦ ◦ . . . ′ ′ v · · n ◦ ... ... ◦ ◦ . . . ′′ ′′ · · v n ... ... ◦ ◦ ◦ Fig. 5.1: Constant distribution  † Remark 5.1.1. Thus, F may be thought of as the Frobenioid associated to 〉 J 〈 divisors on V Z or [i.e., finite formal sums of elements of this set with coefficients in J R ∈ J is constant — that is to say, divisors corresponding j ] whose dependence on . Such constant distributions are depicted in Fig. on V “constant distributions” to J 5.1 above. On the other hand, the product of [underlying categories of] Frobenioids  † may be thought of as a sort of category of “arbitrary distributions” on V F , J J is whose dependence on j ∈ J . arbitrary V i.e., divisors on J Remark 5.1.2. The constructions of Example 5.1 manifestly only require the -ΘNF D † † NF φ of the D -ΘNF-Hodge theater HT . -NF-bridge portion D  In the context of the discussion of Example 5.1, (v), (vi), we note Remark 5.1.3.    † † †  † †   † †  that unlike the case with , , , M M F ,or ,one , M F M M , κ κ × κ sol -sol ∞ ∞   † † [cf. [FrdII], Definition 2.1, (ii)] with cannot perform Kummer theory M F , , mod mod × ×  † or F , F . [That is to say, in more concrete terms, [unlike the case with , M κ sol thecasethatelementsof κ -/ not necessarily κ × -coric rational functions] it is or ∞ ∞ × F -th roots -coric rational functions admit N or ,for N a nonnegative integer!] κ mod  † †   † , F ,or M F implies that perform Kummer theory with can Thefactthatone †  †  π )-action, as well as the “birational monoid ( M D equipped with its natural 1  †  † F F strong rigidity properties , satisfy various or [cf. Corollary 5.3, portions” of (i), below]. For instance, these rigidity properties allow one to recover the additive  † †  ( π D M )-action equipped with its natural of] 0 { on [the union with structure } 1 [cf. the discussion of Example 5.1, (v), (vi)]. That is to say, additive structure — or, equivalently, ring/field structure —on the  † F may only 0 { [the union with of] the “birational monoid portion” of } mod additional datum consisting of the natural berecoveredifoneisgiventhe  †  † F ↪ → [cf. Example 5.1, (iii)]. F embedding mod

132 132 SHINICHI MOCHIZUKI  † without Put another way, if one only considers the category the embedding F mod   † † †  ,then ↪ → F F is subject to a “Kummer black-out” — one con- F mod mod sequence of which is that there is no way to recover the additive structure on the  † F [cf. also Remark 5.1.5 below]. In subsequent “birational monoid portion” of mod   † † †   † discussions, we shall refer to these observations concerning M , F M , , , F sol      †  †   † † †  † † † † , M M , , M M F ,and F M M , by saying that , F , , κ κ × κ mod κ -sol mod ∞ ∞      † † †  † † † M M M , , ,and M M , F , are Kummer-ready ,whereas κ κ × -sol mod κ sol mod ∞ ∞ †  M are Kummer-blind . In particular, the various copies of [and products and κ     † † † † of copies of] , — i.e., F F F F , — considered in Example 5.1, (vii), j J mod 〉 〈 J Kummer-blind are also . Remark 5.1.4. The various functorial reconstruction algorithms for number fields discussed in Example 5.1 are based on the technique of Belyi cuspidaliza- , as applied in the theory of [AbsTopIII], § 1. At a more concrete level, this tion theory of [AbsTopIII], § 1, may be thought of revolving around the point of view that elements of may be expressed geometrically by means of number fields . Belyi maps Moreover, if one thinks of such elements of number fields as “analytic functions” , then the remainder of the theory of [AbsTopIII] [cf., especially, [AbsTopIII], § 5] may be thought of as a sort of theory of “analytic continuation” of the “analytic functions” constituted by ele- number fields in the context of the various maps ments of logarithm . at the various localizations of these number fields This point of view is very much in line with the points of view discussed in Re- marks 4.3.2, 4.3.3. Moreover, the geometric representation of elements of number fields via Belyi maps [i.e., whose existence may be regarded as a reflection of hyperbolic the nature of the projective line minus three points] is reminiscent of — indeed, may perhaps be regarded as an of — the “categories arithmetic analogue hyperbolic analytic continuations ”, i.e., of copies of the of upper half-plane regarded as equipped with their natural hyperbolic metrics , discussed in the “Motivating Example” given in the Introduction to [GeoAnbd]. Here, it is perhaps of interest to note that the inequalities “ ≤ 1” satisfied by the derivatives [i.e., with respect to the respective Poincar ́ e metrics] of the holomorphic maps that appear monotonically in this “Motivating Example” in [GeoAnbd] are reminiscent of the nature of the various “degrees” — i.e., over Q of the decreasing of ramification locus the endomorphisms of the projective line over Q — that appear in the construction of Belyi maps [where we recall that this monotonic decreasing of degrees is the key observation that allows one to obtain Belyi maps which are unramified over the projective line minus three points]. Although the phenomenon of κ -sol -conjugate synchronization Remark 5.1.5. discussed in the final portion of Example 5.1, (i), will not play as central a role in the present series of papers as the conjugate synchronization of local Galois groups that will be discussed in [IUTchII], [IUTchIII], it has the following interesting : consequence

133 ̈ INTER-UNIVERSAL TEICHM 133 ULLER THEORY I Kummer theory The of    κ -sol rat † † †  † † † κ -sol   ) )   M ) π  ”, “ ( M ” ) ”, “ π D ( M ( D D π “( κ 1 1 1 sol mod ∞ × κ -coric functions” , “ F ” ,and “ abstract analogues — i.e., of the of ∞ mod × “ F ” as in Remark 3.1.7, (iii) — that was discussed in Example 5.1, sol (v), may be performed in a fashion that is compatible without any SL †  ) (  D ) of (Aut with the conjugacy indeterminacies poly-action  F . l observation Here, we pause, however, to make the following : At first glance, it may appear as though the analogue obtained by Uchida of the Neukirch-Uchida theorem for maximal solvable quotients of absolute Galois groups of number fields [reviewed, for instance, in [GlSol], § 3] — or, perhaps, some future mono-anabelian version of this result of Uchida — may be applied, in the context of the “ -sol -Kummer κ just discussed, to reconstruct the ring structure theory” on the number fields involved [i.e., in the fashion of Example 5.1, (v)]. In fact, however, “solvable-Uchida-type” meaning- such a reconstruction is, in effect, less from the point of view of the theory of the present series of papers since it is localization opera- fundamentally incompatible with the D -ΘNF-Hodge theater — cf. the tions that occur in the structure of a discussion of Remarks 4.3.1, 4.3.2. Indeed, such a compatibility with localization would imply that the reconstruction of the ring structure may somehow be to the absolute Galois groups of “restricted” contradiction completions at nonarchimedean primes of a number field, i.e., in to the well-known fact that absolute Galois groups of such completions at nonarchimedean primes admit automorphisms that do not arise from field automorphisms [cf., e.g., [NSW], the Closing Remark preceding Theorem 12.2.7]. Finally, we note that this of “solvable-Uchida-type” reconstructions of ring structures with the incompatibility theory of the present series of papers is also interesting in the context of the point of view discussed in Remark 5.1.4: Suppose, for instance, that it was the case that the outer action of the absolute Galois group of a number field on the geometric fun- damental group of a hyperbolic curve over the number field in fact factors through the maximal solvable quotient of the absolute Galois group. Then it is conceivable that some sort of version of the mono-anabelian theory of [AbsTopIII], § ex- 1, for of such a maximal solvable quotient by the geometric fundamental group tensions under consideration could be applied in the theory of the present series of papers to give a reconstruction of the ring structure of a number field that only requires the extensions and is, moreover, use of such with the localization operations compatible that occur in the various types of “Hodge theaters” that appear in the theory of the present series of papers — a state of affairs that would be fundamentally at odds with a quite essential portion of the “spirit” of the theory of the present series of papers, namely, the point of view of dismantling the two underlying “ combinatorial dimensions of a ring ” .Infact,however, the outer action referred to above does not admit such a “solvable fac- torization” . Indeed, the nonexistence of such a “solvable factorization” is a formal consequence of the the Galois injectivity result discussed in [NodNon], Theorem C — a result that depends, in an essential way, on the theory of Belyi maps . Put another way,

134 134 SHINICHI MOCHIZUKI not only play the role of allowing one to perform the sort of Belyi maps [i.e., “arithmetic analytic continuation via Belyi cuspidalizations” discussed in Remark 5.1.4] that is of central importance in the theory of the present series of papers, but also play the role of assuring one that such to the case of “arithmetic analytic continuations” cannot be extended extensions associated to “solvable factorizations” of outer actions of the sort just discussed. Definition 5.2. F -prime-strip ,[relative holomorphic Frobenioid-prime-strip (i) We define a ,or to the given initial Θ-data] to be a collection of data ‡ ‡ F = { F } ∈ v V v non ‡ ‡ ∈ F v is a category ,then C V that satisfies the following conditions: (a) if v v ∼ ‡ C →C [where C is as in Examples which admits an equivalence of categories v v v arc ‡ ‡ ‡ ‡ collection of data F )isa ∈ V C κ , =( D ,then , 3.2, (iii); 3.3, (i)]; (b) if v v v v v consisting of a category, an Aut-holomorphic orbispace, and a Kummer structure ∼ ‡ ) =( C →F , D F ,κ such that there exists an isomorphism of collections of data v v v v v is as in Example 3.4, (i)]. [where F v  ,or F (ii) We define a mono-analytic Frobenioid-prime-strip ,[rel- -prime-strip ative to the given initial Θ-data] to be a collection of data  ‡  ‡ = { F F } v ∈ V v non ‡  that satisfies the following conditions: (a) if ,then v F ∈ V is a split Frobe- v  ‡ C , which admits an isomorphism nioid , whose underlying Frobenioid we denote by v ∼ arc ‡    v F →F F is as in Examples 3.2, (v); 3.3, (i)]; (b) if [where ∈ V ,then v v v   ‡  ‡ is a triple of data, consisting of a Frobenioid C F , an object of TM ,anda v v splitting of the Frobenioid, such that there exists an isomorphism of collections of ∼  ‡   data →F is as in Example 3.4, (ii)]. [where F F v v v  morphism of - (respectively, F (iii) A F -) prime-strips is defined to be a col- , between the various constituent objects of lection of isomorphisms, indexed by V the prime-strips. Following the conventions of § 0, one thus has notions of cap-   F -) and morphisms of capsules of F - (respectively, -) sules of F - (respectively, F . prime-strips  - F (iv) We define a globally realified mono-analytic Frobenioid-prime-strip ,or , [relative to the given initial Θ-data] to be a collection of data prime-strip ∼ ‡  ‡ ‡   ‡  ‡ C C ) Prime( → V , , F =( , { F ρ ) } v V v ∈  ‡ C is a category [which is, in fact, that satisfies the following conditions: (a)  of equipped with a Frobenioid structure] that is isomorphic to the category C mod

135 ̈ INTER-UNIVERSAL TEICHM 135 ULLER THEORY I − Example 3.5, (i); (b) “Prime( )” is defined as in the discussion of Example 3.5, ∼  ‡  ‡  ‡  (i); (c) Prime( = { C F → V } is a bijection of sets; (d) is an -prime- F F ) V v ∈ v ∼ rlf ‡ rlf Φ ” are defined as in the ,where“Φ → ρ :Φ ”and“Φ strip; (e) ‡ ‡     ‡ ‡ v ,v C C ,v C C v v discussion of Example 3.5, (i), is an isomorphism of topological monoids [both of R which are, in fact, isomorphic to ]; (f) the collection of data in the above display ≥ 0  of Example 3.5, (ii). A morphism of F is isomorphic to the collection of data mod  -prime-strips is defined to be an isomorphism between collections of data as F 0, one thus has notions of capsules discussed above. Following the conventions of §   . and -prime-strips F -prime-strips morphisms of capsules of F of non ‡ ‡ ∈ D D } for V ∈ be a D -prime-strip , v = V { .Write v (v) Let V ∈ mod w w . Then [cf. the discussion of Example 5.1, (i); Re- the valuation determined by v ‡ D ), in a functorial ( from group-theoretically mark 3.1.7, (i)] one may construct π 1 v ” [cf. the algorithms of [AbsTopII], corresponding to “ C fashion, a profinite group v Corollary 3.3, (i), which are applicable in light of [AbsTopI], Example 4.8; [Ab- ‡ ‡ ( )asan open subgroup ;wewrite D D sTopIII], Theorem 1.9], which contains π v v 1 0 ( − for B of this profinite group, so we obtain a natural morphism ) ‡ ‡ → D D v v — i.e., a “category-theoretic version” of the natural morphism of hyperbolic orbi- good bad V = X ∈ × v K if → C C if v ∈ V . ,or X → K × = X X curves K v v v v K − − → → K v K v Belyi cuspidal- Next, let us observe [cf. Remark 3.1.7, (i); the construction of the × “ K field of [AbsTopIII], Theorem 1.9, (a), and of the izations ” of [AbsTopIII], Z NF ‡ π Theorem 1.9, (d), (e)] that one may construct group-theoretically from D ), in ( 1 v a functorial fashion, an isomorph rat ‡ ‡ ( )(  π ( )) D D π 1 v v 1 ‡ ( D ) of the π [i.e., equipped with its natural surjection to ́ etale fundamental group v 1 and well-defined up to inner automorphisms determined by elements of the kernel ) the of this natural surjection] of the scheme obtained by base-changing to ( F v mod . Next, let us recall [cf. [AbsTopIII], Corollary 1.10, (b), (c), generic point of C F mod ′ ‡ (d), (d from ( group-theoretically D ), in a functorial π )] that one may construct 1 v  [which is naturally isomorphic to O fashion, an ind-topological monoid ] F v ‡ ) ( D M v v ‡ D )-action, as well as isomorphs of the pseudo- ( π equipped with its natural 1 v κ [cf. the -, and C κ × -coric rational functions associated to κ monoids of -, v ∞ ∞ discussion of Remark 3.1.7, (i), (ii); [AbsTopII], Corollary 3.3, (iii), which is appli- cable in light of [AbsTopI], Example 4.8; [AbsTopIII], Theorem 1.9, (a), (d), (e); ′ )] — which we shall denote [AbsTopIII], Corollary 1.10, (d), (d ‡ ‡ ‡ M D ) ) , M ( D ( ( D , ) M v × v κv κ κv v v ∞ ∞ rat ‡ ‡ .Thus, )maybeiden- ( D ) -actions ( M D natural π — equipped with their κv v v 1 rat ‡ ‡ D ( ), and [if we use the D ( ) M of -invariants tified with the subset of π κv v v 1 ∞

136 136 SHINICHI MOCHIZUKI × ” to denote the subset of invertible elements of a pseudo-monoid, superscript “ ‡ × ‡ × may be identified with M ( D ) D ) ( . M then] v × v v v κ ∞ ‡ (vi) We continue to use the notation of (v). Suppose further that F = ‡ whose associated F [cf. Remark 5.2.1, -prime-strip is an F -prime-strip } D { V ∈ w w ‡ ‡ { .Let D D } = (i), below] is equal to ∈ w V w ‡ ‡ π  ( M ) D v v 1 ‡ D ind-topological monoid equipped with a continuous action by ( )that be an π v 1 is isomorphic [i.e., as an ind-topological monoid equipped with a continuous action ‡ ‡ ‡ D )] to the pair ) constructed in (v). One may regard ( D π D ) M ( ( by π v v v 1 v 1 ‡ ‡ ‡ [cf. (i), (a)] D related to the Frobenioid )  F M as being ( π such a pair v v v 1 ‡ D = unique isomorphism of corresponding to the via the identity automorphism ‡ ‡ -adic Frobenioid p and the [cf. Corollary 5.3, (ii), below] between D F } { V ∈ v w w v determined [cf. Remark 3.3.2] by the pair ‡ ‡ D  ) ( M π 1 v v ‡ ‡ ( obtained by restricting D theactionofthepair )  π M open subgroup to the v 1 v ‡ ‡ ( D ) ⊆ π pseudo- ( ) [cf. (v)]. We shall refer to a pair [i.e., consisting of a D π v 1 v 1 rat ‡ D )] ( π equipped with a continuous action by monoid v 1 rat ‡ rat ‡ ‡ ‡ M D M ) (respectively, π  ( ) D )  ( π κv v v v × κ 1 1 ∞ ∞ ‡ on -coric × -coric ) structure κ κ F if it is isomorphic [i.e., (respectively, as an ∞ ∞ v equipped with a continuous action by pseudo-monoid as a pair consisting of a ‡ rat )] to the pair D ( π v 1 rat ‡ ‡ rat ‡ ‡ π ( ) D D ) (respectively, π ( ( M D ) M D ( )) κv v κ × v v v v 1 1 ∞ ∞ ) κ -coric (respectively, -coric κ × of (v). Suppose that we have been given such an ∞ ∞ ‡ F . In the following, we shall use the notational convention “ μ ))” − (( on structure v ̂ Z Θ ‡ π ( D ( )) for the cyclotome μ introduced in Example 5.1, (v). Also, let us write v 1 ̂ Z μ “ “via the (Π )” of [AbsTopIII], Theorem 1.9, which we think of as being applied − ( ) ̂ Z ‡ ‡ that ( ( ) D M ). Then let us observe D -approach” π [cf. Remark 3.1.2] to Θ v κv v 1 ∞ ‡ [cf. [AbsTopIII], Theorem 1.9, constructed )) is, in effect, D ( (respectively, M v v κ × ∞ of (d)] as a subset ‡ 1 Θ ( ( μ D H ))) ( π H, lim 1 v → − ̂ Z H rat ‡ ( D ). On the other hand, ranges over the open subgroups of π —where H v 1 rat ‡ )on N D ( -th roots of π consideration of [i.e., of the action of Kummer classes v 1 ‡ (respectively, M of natural injection N elements, for positive integers ] yields a κv ∞ ‡ M )into κ × v ∞ ‡ 1 ‡ 1 H ))) ( H, μ μ H, ( M M ( H )) (respectively, lim ( lim v κv × κ ∞ ∞ H H → − − → ̂ ̂ Z Z ‡ rat D ), and we observe that the ( ranges over the open subgroups of H —where π v 1 asserted injectivity follows immediately from the corresponding injectivity in the

137 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 137 ‡ ‡ ( case of ) (respectively, M )). In particular, it follows imme- D M ( D v κ v v κv × ∞ ∞ diately [cf. the discussion of Example 5.1, (v)], by considering divisors of zeroes -coric function ” given in Remark 3.1.7, (i)] and poles [cf. the definition of a “ κ associated to Kummer classes of rational functions as in [AbsTopIII], Proposition 1.6, (iii), from the elementary observation that, relative to the natural inclusion ̂ Q ⊗ Q , → ↪ Z ⋂ × ̂ Z } = { 1 Q 0 > that there exists a unique isomorphism of cyclotomes ∼ ∼ ‡ ‡ ‡ Θ Θ ‡ ( )) )) → μ ( ( π M μ → ) (respectively, μ ( M )) ( π D ( D μ 1 × v κv κ v 1 v ∞ ∞ ̂ ̂ Z Z ̂ ̂ Z Z such that the resulting isomorphism between direct limits of cohomology modules isomorphism as considered above induces an ∼ ∼ ‡ ‡ ‡ ‡ M ) (respectively, M ) ( ( → D ) M → D M v v κ κv v κv × v κ × ∞ ∞ ∞ ∞ rat ‡ ( D )]. In a similar equipped with continuous actions by π [i.e., of pseudo-monoids v 1 vein, it follows immediately from the theory summarized in [AbsTopIII], Corollary ′ ), that there exists a unique isomorphism of cyclotomes 1.10, (d), (d ∼ Θ ‡ ‡ μ ( ) D π )) ( → μ ( M v 1 v ̂ Z ̂ Z such that the resulting isomorphism between direct limits of cohomology modules induces an isomorphism ∼ ‡ ‡ M D M ( ) → v v v ‡ [i.e., of monoids equipped with continuous actions by D π )]. In particular, it ( 1 v follows immediately from the above discussion that ‡ always admits an F κ -coric (respectively, ,which κ × -coric ) structure ∞ v ∞ unique up to a uniquely determined isomorphism [i.e., of is, moreover, rat ‡ D )]. ( π equipped with continuous actions by pseudo-monoids v 1 Thus, in the following, we shall regard, without further notice , this uniquely deter- ‡ κ -coric (respectively, F κ × -coric ) structure on as a collection of data mined ∞ v ∞ ‡ . Here, we observe that the various isomorphisms F naturally associated to that is v ‡ ‡ , M M of the last few displays allow one to regard the pseudo-monoids × v κ κv ∞ ∞ ‡ ‡ related to the Frobenioid F via M as being [cf. the discussion at the begin- v v ‡ ‡ M and F ]andthe ning of the present (vi) concerning the relationship between v v morphisms ∼ × ‡ ‡ × ‡ ‡ × ‡ ↪ → M M → , M M M → ↪ κ κv × v v κv × κ v ∞ ∞ ∞ ∞ induced by the various isomorphisms of the last few displays, together with the corresponding inclusions/equalities ‡ ‡ D ( ) D , ) ⊆ M ( M κ v v κv v × ∞ ∞ ‡ × ‡ × ‡ × ) M ⊆ ( ) ( D D ) = M ( D M v v κv v κ × v v ∞ ∞

138 138 SHINICHI MOCHIZUKI [cf. the discussion at the end of (v)]. Also, we shall write ‡ ‡ M M ⊆ κv κv ∞ rat ‡ π D -invariants ) for the “sub-pseudo-monoid” of . In this context, we observe ( v 1 further that it follows immediately from the discussion of Remark 3.1.7, (i), (ii) [cf. also [AbsTopII], Corollary 3.3, (iii), which is applicable in light of [AbsTopI], Ex- ample 4.8], and the theory summarized in [AbsTopIII], Theorem 1.9 [cf., especially, [AbsTopIII], Theorem 1.9, (a), (d), (e)], and [AbsTopIII], Corollary 1.10, (h), that rat ‡ ‡ ‡ on determines ( κ D F )  × M -coric structure π any × v v v ∞ κ 1 ∞ -coric structure κ an associated ∞ ‡ ‡ rat ‡ π M M D ( ⊆  ) v κv v κ × 1 ∞ ∞ of the restriction by considering the subset of elements for which the associated Kummer class [as in the above discussion] to some [or, equiv- rat ‡ ( ) that cor- D alently, every — cf. Remark 5.2.3 below] subgroup of π v 1 responds to an open subgroup of the decomposition group of some ∼ ‡ × → determines a torsion element ∈ M strictly critical C point of v v × ‡ M [i.e., corresponds to a root of unity], × v κ ∞ and, moreover, that the operation of restricting Kummer classes [as in the above discus- ‡ ‡ rat ‡ sion] arising from ⊆ M to subgroups of π )thatcorre- D ( M κv v κv 1 ∞ ) -valued points spond to F decomposition groups of non-critical ( v mod submonoid the yields a functorial algorithm for reconstructing of C v gp ‡ ‡ [where the superscript “gp” denotes the D M ) -invariants of ( π of v 1 v ind-topological field structure on the groupification], together with the ‡ M κ -coric structure ,fromthe union of this monoid with { 0 } ∞ κv ∞ ‡ associated to F . v A similar statement to the statement of the last display holds, if one replaces the F phrase “( ” by the phrase “ -valued points ) F ” and the phrase -valued points v mod v ‡ ‡ gp gp ‡ ‡ π ”. of D M ( ) ” by the phrase “ pair -invariants ( D M )  submonoid of π “ v 1 1 v v v arc ‡ ‡ (vii) Let D D for the } V ∈ v be a D = , v ∈ V { .Write -prime-strip V ∈ w mod w . Then [cf. the discussion of Example 5.1, (i); Remark valuation determined by v ‡ D ,in algorithmically -holomorphic space from the Aut 3.1.7, (i)] one may construct v ‡ C D corresponding to “ ”[cf. -holomorphic orbispace a functorial fashion, an Aut v v natural morphism the algorithms of [AbsTopIII], Corollary 2.7, (a)], together with a ‡ ‡ D → D v v — i.e., an “Aut-holomorphic orbispace version” of the natural morphism of hy- def perbolic orbicurves X → = . Here, we observe [cf. × K C × K X K v v v F ) ( v mod − → − → K v is a K -core ; [AbsTopIII], Corollary 2.3] that one has a natural the fact that C K isomorphism ∼ ‡ Aut( Z ) ) → Gal( K 2 / ( F / ) Z )( ↪ → D v mod v v

139 ̈ INTER-UNIVERSAL TEICHM 139 ULLER THEORY I ‡ holomorphic D is — i.e., obtained by considering whether an automorphism of v or anti-holomorphic — from the group of automorphisms of the Aut-holomorphic ‡ onto the Galois group Gal( K D / ( F ).Write ) orbispace v v v mod ‡ rat ‡ D → D v v universal for the projective system of Aut-holomorphic orbispaces that arise as ‡ D [i.e., open sub-orbispaces “co-finite” open sub-orbispaces of covering spaces of v determined by forming complements of finite sets of points of the underlying topo- ‡ ]thatcontain D every strictly critical point [cf. Remark 3.1.7, logical orbispace of v (i)], as well as every point that is not an NF-point [cf. Remark 3.4.3, (ii)], of rat ‡ ‡ ‡ deck transformations is D up to the action of .Thus, over D D well-defined v v v ‡ NF-points of [cf. the D countability ; the discussion of compatible of the set of v systems of basepoints at the end of Remark 2.5.3, (i)]. Next, let us recall the A [cf. the discussion of Example 3.4, (i), complex archimedean topological field ‡ D v as well as Definition 3.6, (b); the discussion of (i) of the present Definition 5.2]. ∼ ) of the topological field Z 2 / Z ) for the group of automorphisms ( A Write Aut( ‡ = D v in [Ab- . Observe that it follows immediately from the construction of A A ‡ ‡ D D v v -holomorphic A Aut natural is equipped with a sTopIII], Corollary 2.7, (e), that ‡ D v tautological co- structure [cf. [AbsTopIII], Definition 4.1, (i)], as well as with a [cf. [AbsTopIII], Definition 2.1, (iv); [AbsTopIII], Proposition holomorphicization ‡ D .Write 2.6, (a)] with v ‡ ( A D ⊆ ) M ‡ v v D v for the topological submonoid consisting of nonzero elements of norm ≤ 1 [i.e.,  “ O ”]. Thus, A of the groupification } 0 may be identified with the union with { ‡ D C v ‡ gp pseudo-monoids of D × ) -coric rational . Moreover, the ( κ -, κ κ -, and M ∞ ∞ v v [cf. the discussion of Remark 3.1.7, (i), (ii)] may be associated to C functions v algorithmic constructions represented, via [cf. [AbsTopIII], Corollary 2.7, (b)], as pseudo-monoids of “meromorphic functions” ‡ ‡ ‡ ) ) , M ) D ( D D ( ( , M M κv × v κv v κ v v ∞ ∞ — i.e., as sets of morphisms of Aut-holomorphic orbispaces from [some constituent ‡ gp rat ‡ that are to M D tauto- ) D ( compatible with the of the projective system] v v v logical co-holomorphicization just discussed and, moreover, satisfy conditions corre- ‡ ( D ) M sponding to the conditions of the final display of Remark 3.1.7, (i). Here, κv v ‡ descend )that ( to some D M may be identified with the subset of elements of v κv ∞ ‡ D and, moreover, are equivariant with respect to co-finite open sub-orbispace of v ‡ A D ) ↪ → Aut( ); [if we use the superscript “ ”tode- × the unique embedding Aut( ‡ v D v ‡ × ( D may be ) M note the subset of invertible elements of a pseudo-monoid, then] v v × ‡ × ‡ × ‡ and ( ; we observe that both M ( D ) ) M D D ( ) M identified with × κ κ × v v v v v v ∞ ∞ × 1 [i.e., “ O are isomorphic, as abstract topological monoids, to ”]. S C ‡ F = (viii) We continue to use the notation of (vii). Suppose further that ‡ D } [cf. Remark 5.2.1, -prime-strip F is an F -prime-strip whose associated { V w w ∈ ‡ ‡ } { D D .Write = (i), below] is equal to w ∈ V w ‡ M v

140 140 SHINICHI MOCHIZUKI  ‡ [i.e., “ O )” — cf. the discussion of Example 3.4, (i); topological monoid for the ( C v Kummer structure of the domain Definition 3.6, (b)] that appears as the portion ‡ F [cf. (i) of the present Definition 5.2]. Thus, the of the data that constitutes v ‡ mayberegardedasan isomorphism of F portion of Kummer structure v topological monoids ∼ ‡ ‡ ) → D M ( M v v v  O [both of which are abstractly isomorphic to ]. In particular, the Kummer struc- C ∼ ‡ gp ‡ gp ( → [both M D ) M ture determines an isomorphism of topological groups v v v × of which are abstractly isomorphic to C A ], hence also a of Aut( natural action ) ‡ D v ‡ gp M . Next, let us observe that the pseudo-monoid of -) κ - (respectively, × κ on ∞ ∞ v [cf. the discussion of Remark 3.1.7, (i), coric rational functions associated to C v algorithmic constructions [cf. [AbsTopIII], Corollary (ii)] may be represented, via “meromorphic functions” 2.7, (b)], as the pseudo-monoid of ‡ ‡ M M ) (respectively, κv κ × v ∞ ∞ by considering the set of maps from [some constituent of the projective system] ‡ rat to D v ‡ gp M v that satisfy the following condition: the map from [some constituent of the pro- ‡ rat gp ‡ composing M ( to D D ) the given map with the obtained by jective system] v v v inverse of [the result of applying “gp” to] the Kummer structure isomorphism ∼ ‡ ‡ ‡ D ) ( → )(re- M D determines an element of the pseudo-monoid M ( M κv v v v v ∞ ‡ D )) discussed in (vii) above. We shall refer to ( M spectively, v κ v × ∞ ‡ ‡ ) M (respectively, M κ × v κv ∞ ∞ ‡ as the [uniquely determined] κ -coric (respectively, F κ × -coric ) structure on ∞ ∞ v and write ‡ ‡ M M ⊆ κv κv ∞ ‡ descend to some co-finite open sub-orbispace of D for the subset of elements that v ‡ equivariant D ) ↪ → and, moreover, are with respect to the unique embedding Aut( v Aut( ))” − (( ). In the following, we shall use the notational convention “ μ A ‡ D v ̂ Z introduced in Example 5.1, (v). Also, let us write def gp ‡ Θ ‡ / ) ) = Hom( ( Q ) Z , M D ( D μ v v v ̂ Z × ‡ μ ‡ , Z M / Q = Hom( D ) = Hom( Q / Z ( M ) ( , D ) ) v v v v ‡ ‡ × ‡ of D ) ( ⊆ M ); ( D D ( ) denotes the topological group of units M M —where v v v v v v ‡ μ × ‡ D denotes the subgroup of ; we observe that torsion elements ⊆ M ( ( ) D ) M v v v v discussed above induces a natural “Kummer the Kummer structure isomorphism ∼ Θ ‡ ‡ M ) [of profinite groups D μ ) ( → ( μ structure cyclotomic isomorphism” v v ̂ Z ̂ Z ̂ Z ]; the superscript “Θ” may be thought of as expressing abstractly isomorphic to Θ μ the fact that we wish to apply to “ ( − )” the interpretation via the archimedean ̂ Z version of the Θ -approach , i.e., the interpretation in terms of cuspidal inertia

141 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 141 , discussed in Remark 3.4.3, (i). In this context, we observe that these groups cuspidal inertia groups may be interpreted as profinite completions of of subgroups the group of deck transformations rat ‡ D ) ( π v 1 determined, up to inner automorphism, by the projective system of covering spaces ‡ rat ‡ ‡ ‡ . Here, we observe that the pseudo-monoids M → M ( D D D ), v κv κv v v ∞ ∞ ‡ ‡ rat ‡ M -actions ( (respectively, )admit natural π D in such a ( ), D ) M × v × v v v κ κ 1 ∞ ∞ pairs way that each of the rat ‡ ‡ ‡ rat ‡ ( M ) D ), π ( ( D D )  M π κv v κv v v 1 1 ∞ ∞ rat ‡ ‡ ‡ rat ‡ ) ) M M π  ) ( ( D D )), π (respectively, D ( × κ v v κ × v v v 1 1 ∞ ∞ ‡ rat ( D ) -conjugacy . Next, let us observe that by consider- is well-defined up to π v 1 just discussed on elements of cuspidal inertia groups ing the action of the various ‡ ‡ M M ) — i.e., in effect, by consider- (respectively, pseudo-monoid the κv κ × v ∞ ∞ “local Kummer classes” ing, in the fashion of [AbsTopIII], Proposition 1.6, (iii), atthepointsthatgiverisetothesecuspidalinertiagroups—weobtainvari- -multiples — i.e., corresponding to the ous at the point Q order of zeroes or poles that gives rise to the cuspidal inertia group under consideration — of the Kummer ∼ Θ ‡ ‡ M ( D ) discussed above. In par- ) ( → μ μ structure cyclotomic isomorphism v v ̂ Z ̂ Z natural identification [cf. the various definitions involved!] ticular, relative to the ‡ ‡ ‡ μ of )with M )), it follows immediately M μ ( μ ( ( M ) (respectively, κv v v × κ ∞ ∞ ̂ ̂ ̂ Z Z Z [cf. the discussion of Example 5.1, (v)], by considering [i.e., in the fashion just dis- cussed] -coric function ”given divisors of zeroes and poles [cf. the definition of a “ κ in Remark 3.1.7, (i)] of meromorphic functions, from the elementary observation ̂ → Z that, relative to the natural inclusion ⊗ Q , Q ↪ ⋂ × ̂ Z Q { = 1 } 0 > the Kummer structure cyclotomic that one may algorithmically reconstruct isomorphism ∼ ∼ Θ ‡ ‡ Θ ‡ ‡ μ ) μ ) )) ( M M ( μ ) (respectively, μ D → → ( ( D v v × κv κ v ∞ ∞ ̂ ̂ Z Z ̂ ̂ Z Z ∼ μ ‡ ‡ μ → D M ) ( Kummer structure isomorphism — hence also the M v v κv ∞ ∼ μ ‡ ‡ μ M ’s” denote the ( → μ M D (respectively, ) ) [where the superscript “ v v κ × v ∞ subgroups of torsion elements ]—from the projective system of coverings of Aut -holomorphic orbispaces ‡ ‡ rat rat ‡ D D , together with the abstract pseudo-monoid with π D - ( → ) v v v 1 rat ‡ ‡ ‡ rat ‡ D  )  π M ) D (respectively, ). ( M ( action π κv v v κ × v 1 1 ∞ ∞ Since, moreover, a rational algebraic function is completely determined by its divisor of zeroes and poles together with its value at a single point , we thus conclude that one may algorithmically reconstruct the isomorphism(s) of pseudo-monoids

142 142 SHINICHI MOCHIZUKI ‡ Kummer structure F on [i.e., by the Kummer structure determined by the v ∼ ‡ ‡ M isomorphism discussed above] ) ( M D → v v v ∼ ∼ ‡ ‡ ‡ ‡ → ) M ) , M M → ( ( D D M v κv v κv κv κv ∞ ∞ ∼ ‡ ‡ M ( M D ) ) (respectively, → v × v κ × v κ ∞ ∞ from the projective system of coverings of Aut -holomorphic orbispaces ‡ rat ‡ rat ‡ D D , together with the abstract pseudo-monoid with π → - ( ) D v v v 1 rat ‡ ‡ ‡ rat ‡ )  )and M (  (respectively, π D ) ( M D action π v v κv κ × v 1 1 ∞ ∞ the collection of splittings × ‡ ‡ ‡ ‡ μ M  (respectively, M M M  ) × v κv κ κv κ × v ∞ ∞ ∞ ∞ — where the superscript “ × ”) denotes the subgroup of ” (respectively, “ μ units , which con- torsion elements (respectively, the topological group of ‡ M dense tains the subgroup of torsion elements as a subgroup) of κv ∞ ‡ M (respectively, ) — determined [and parametrized], via the oper- κ × v ∞ ation of restriction , by the collection of systems of strictly critical ‡ rat ‡ → D D [i.e., systems of points lying over some strictly of points v v ‡ ]. D critical point of v In this context, we observe further that it follows immediately from the discussion of Remark 3.1.7, (ii) [cf. also [AbsTopIII], Corollary 2.7, (b)], that the κ -coric structure ∞ ‡ ‡ ⊆ M M × v κv κ ∞ ∞ ‡ ‡ κ × -coric structure on M F may be constructed from the v ∞ κ × v ∞ ‡ by considering the subset of elements for which the restriction on F v of system of strictly critical points to some [or, equivalently, every] ∼ × ‡ ‡ rat ‡ ‡ × → M D D [i.e., corresponds → is a M torsion element ∈ v v v × κ v ∞ to a root of unity], and, moreover, that ‡ ‡ to systems M M ⊆ restricting the operation of elements of κv κv ∞ ‡ rat ‡ that lie over Aut( points D D ) -invariant non-critical of points of v v ‡ reconstructing the submonoid D yields a functorial algorithm for of v gp ‡ ‡ [where the superscript “gp” denotes the D M ) -invariants of of Aut( v v groupification], together with the topological field structure on the union ‡ associated -coric structure κ M ,fromthe } 0 { of this monoid with ∞ κv ∞ ‡ F . to v A similar statement to the statement of the last display holds if one replaces the ‡ D submonoid ) -invariant ” by the phrase “ arbitrary ” and the phrase “ phrase “Aut( v ‡ gp gp ‡ ‡ -invariants of M D ) Aut( of ” by the phrase “ monoid M ”. v v v

143 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 143 Remark 5.2.1. (i) Note that it follows immediately from Definitions 4.1, (i), (iii); 5.2, (i), functorial (ii); Examples 3.2, (vi), (c), (d); 3.3, (iii), (b), (c), that there exists a  - (respectively, -) prime-strips from F - (respectively, algorithm for constructing D D  prime-strips . -) F (ii) In a similar vein, it follows immediately from Definition 5.2, (i), (ii); Exam- ples 3.2, (vi), (f); 3.3, (iii), (e); 3.4, (i), (ii), that there exists a functorial algorithm ‡ ‡  ‡  -prime-strip { } an F F = F F -prime-strip F for constructing from an v ∈ V v ‡ ‡  ‡  F { F → F = }  v ∈ V v ‡ of mono-analyticization . Next, let us recall from — which we shall refer to as the F of the category the discussion of Example 3.5, (i), the relatively simple structure  ”, i.e., which may be summarized, roughly speaking, as a collection, indexed C “ mod ∼ V by R , of copies of the topological monoid → V , which are related to one 0 mod ≥ another by a “product formula”. In particular, it follows immediately [cf. Definition rigidity of the associated to the Frobenioids that 5.2, (i)] from the divisor monoids [cf., especially, ∈ V of an F -prime-strip v appear at each of the components at topological field structure of the field “ the A ” of Example 3.4, (i)!] that one D v ‡ functorial algorithm [cf. the F , via a -prime-strip may also construct from the F constructions of Example 3.5, (i), (ii)], a collection of data def ∼ ‡ ‡ ‡  ‡   ‡  ‡ , Prime( F C  ) → → V , =( F C , { F ρ ) } v ∈ v V — i.e., consisting of a category [which is, in fact, equipped with a Frobenioid ‡   -prime-strip F , and an isomorphism of topological structure], a bijection, the F ‡ ‡   , respectively, at each C F v ∈ V — which is iso- and monoids associated to  of Example 3.5, (ii), i.e., which forms an F morphic to the collection of data mod  F [cf. Definition 5.2, (iv)]. -prime-strip Remark 5.2.2. Thus, from the point of view of the discussion of Remark 5.1.3, non ∈ V — cf. the theory of [FrdII], § 2], are Kummer-ready F v -prime-strips [i.e., at  Kummer-blind are -prime-strips . F whereas κ -coric structures from In the context of the construction of Remark 5.2.3. ∞ κ × -coric structures in Definition 5.2, (vi), we make the following observation. ∞ bad ∈ V , it is natural to take the corresponding to decomposition groups v When strictly critical points [i.e., to which one restricts the Kummer classes under con- that C sideration] to be decomposition groups that correspond to the point of v arises as the image of the zero-labeled evaluation points [i.e., evaluation points | — cf. the discussion of Example 4.4, (i)]. In the corresponding to the label 0 ∈| F l may also be described simply as the C notation of Example 4.4, (i), this point of v ”. μ point that arises as the image of the point “ − Corollary 5.3. (Isomorphisms of Global Frobenioids, Frobenioid-Prime- Strips, and Tempered Frobenioids) Relative to a fixed collection of initial Θ -data :

144 144 SHINICHI MOCHIZUKI i  i  F , (respectively, 2 F ,let )bea category which is equiv- i (i) For =1  †  i  † F F F ) of Example 5.1, (iii). Thus, (respectively, alent to the category  i natural Frobenioid structure [cf. [FrdI], F ) is equipped with a (respectively, Corollary 4.11; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper]. Write i  i  Base( ) F F ) ) for the base category of this Frobenioid. Then (respectively, Base( the natural map 2  1  1 2   Isom( , F F ) F Base( , F ) )) → Isom(Base( 1   2 2   1 Isom( (respectively, Isom(Base( F ) ) , Base( F F , )) ) → F [cf. [FrdI], Corollary 4.11; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present bijective paper] is . i i =1 , i ,let (ii) For 2 F -prime-strip ; D D the F -prime-strip associated be an i [cf. Remark 5.2.1, (i)]. Then the natural map F to 1 2 1 2 Isom( ) → Isom( F D , , D ) F bijective . [cf. Remark 5.2.1, (i)] is   i   i D F the be an F -prime-strip -prime-strip ; D ,let 2 =1 i (iii) For , i  [cf. Remark 5.2.1, (i)]. Then the natural map F associated to 1  2  1  2  Isom( D F , F ) D → ) Isom( , surjective . [cf. Remark 5.2.1, (i)] is bad (iv) Let v F . Recall the category ∈ V of Example 3.2, (i). Thus, F v v natural Frobenioid structure [cf. [FrdI], Corollary 4.11; is equipped with a . Then the natural homomorphism [EtTh], Proposition 5.1], with base category D v ) [cf. Example 3.2, (vi), (d)] is ) → Aut( D . bijective F Aut( v v Proof. category-theoreticity of the “iso- Assertion (i) follows immediately from the ∼  †   † M morphism of Example 5.1, (v) [cf. also the surrounding → D M ( ” ) discussion; Example 5.1, (vi)]. [Here, we note in passing that this argument is en- ∼   Th tirely similar to the technique applied to the proof of the equivalence “ EA → ” T of [AbsTopIII], Corollary 5.2, (iv).] Assertion (ii) (respectively, (iii)) follows imme- diately from [AbsTopIII], Proposition 3.2, (iv); [AbsTopIII], Proposition 4.2, (i) [cf. also [AbsTopIII], Remarks 3.1.1, 4.1.1; the discussion of Definition 5.2, (vi), (viii), of the present paper] (respectively, [AbsTopIII], Proposition 5.8, (ii), (v)). Finally, we consider assertion (iv). First, we recall that since automorphisms temp 0 [cf. B ( X X ) = necessarily arise from automorphisms of the scheme of D v v v surjectivity [AbsTopIII], Theorem 1.9; [AbsTopIII], Remark 1.9.1], follows immedi- .Tothis injectivity . Thus, it remains to verify F ately from the construction of v )). For simplicity, we suppose [without loss of ) → Aut( D end, let α ∈ Ker(Aut( F v v α lies over the identity self-equivalence of D generality] that .ThenI claim that v ,it α is [isomorphic to — cf. § 0] the identity self-equivalence of F to show that v suffices to verify that α induces [cf. [FrdI], Corollary 4.11; [EtTh], Proposition 5.1] the identity on the rational function and divisor monoids of F . v

145 ̈ INTER-UNIVERSAL TEICHM 145 ULLER THEORY I F Indeed, recall that since [cf. [EtTh], Definition 3.6, is a Frobenioid of model type v preserves α base-Frobenius (ii)], it follows from [FrdI], Corollary 5.7, (i), (iv), that induces the identity on the rational function pairs . Thus, once one shows that α F of divisor monoids and , it follows, by arguing as in the construction of the v equivalence of categories given in the proof of [FrdI], Theorem 5.2, (iv), that the obtained in [FrdI], Proposition 5.6, determine [cf. Remark 5.3.3 below; various units the argument of the first paragraph of the proof of [FrdI], Proposition 5.6] an between and the identity self-equivalence of F α isomorphism , as desired. v induces the identity on the rational func- Thus, we proceed to show that α [cf. , as follows. In light of the category-theoreticity F and tion of divisor monoids v cyclotomic rigidity isomorphism [EtTh], Theorem 5.6] of the of [EtTh], Proposition identity on the rational function monoid follows α 5.5, the fact that induces the naturality of the Kummer map [cf. the discussion of Remark immediately from the by [EtTh], Proposition 3.2, 3.2.4; [FrdII], Definition 2.1, (ii)], which is injective (iii) — cf. the argument of [EtTh], Theorem 5.7, applied to verify the category- theoreticity of the Frobenioid-theoretic theta function. Next, we consider the effect cus- . To this end, let us first recall that α preserves on the divisor monoid of of α F v and elements of the monoids that appear in this divisor monoid pidal non-cuspidal [cf. Remark 3.2.4, (vi); [EtTh], Proposition 5.3, (i)]. In particular, by considering the non-cuspidal portion of the divisor of the Frobenioid-theoretic theta function α ,since α has already been shown and its conjugates [each of which is preserved by α ], we conclude that F to induce the identity on the rational function monoid of v identity on the elements of the monoids that appear in induces the non-cuspidal F the divisor monoid of [cf. [EtTh], Proposition 5.3, (v), (vi), for a discussion v of closely related facts]. In a similar vein, since any divisor of degree zero on an of the elliptic curve admits a elliptic curve that is supported on the torsion points principal positive multiple which is , it follows by considering the cuspidal portions of divisors of appropriate rational functions [each of which is preserved by α ,since α has already been shown to induce the identity on the rational function monoid of elements of the monoids that ]that α also induces the identity on the cuspidal F v . This completes the proof of assertion (iv). © F appear in the divisor monoid of v Remark 5.3.1. (i) In the situation of Corollary 5.3, (ii), let 1 2 φ : → D D be a morphism of D -prime-strips [i.e., which is not necessarily an isomorphism!] that induces an isomorphism between the respective collections of data indexed by ∼ good 1   2  v ,aswellasan V isomorphism φ → ∈ D : between the associated D  -prime-strips that by applying [cf. Definition 4.1, (iv)]. Then let us observe D φ lifts to a uniquely determined “arrow” Corollary 5.3, (ii), it follows that 1 2 ψ : → F F —whichwethinkofas “lying over” φ — defined as follows: First, let us recall that, in light of our assumptions on φ , it follows immediately from the construction

146 146 SHINICHI MOCHIZUKI -adic and p [cf. Examples 3.2, (iii); 3.3, (i); 3.4, (i)] of the various archimedean [cf. [FrdII], Example 1.1, (ii); [FrdII], Example 3.3, (ii)] that appear Frobenioids F -prime-strip that it makes sense to speak of the in an — i.e., by “pull-back” 0; [FrdI], Proposition 1.6] — [cf. [FrdI], “categorical fiber product” forming the § 2 F via the various morphisms -prime-strip of the Frobenioids that appear in the F ∈ V that constitute φ . That is to say, it follows from our assumptions on at v determines a pulled-back F - [cf. also [AbsTopIII], Proposition 3.2, (iv)] that φ φ ∗ 2 ( -prime-strip [cf. Remark 5.2.1, (i)] is D F )”, whose associated φ prime-strip “ 1 D . On the other hand, by Corollary 5.3, (ii), it follows tautologically equal to -prime-strips uniquely determines that this tautological equality of associated D an ∼ 2 2 ∗ 1 1 F → ). Then we φ the arrow ψ : F F → ( F to be the define isomorphism ∼ 1 ∗ 2 collection of data consisting of φ and this isomorphism F ( → F ); we refer to ψ φ “morphism uniquely determined by as the or the “uniquely determined morphism φ ” ” that lies over . Also, we shall apply various terms used to φ describe a morphism -prime-strips to the “arrow” of D -prime-strips determined by φ . of φ F - (ii) The conventions discussed in (i) concerning liftings of morphisms of D . We leave the routine details prime-strips may also be applied to poly-morphisms to the reader. Just as in the case of Corollary 5.3, (i), (ii), the rigidity property Remark 5.3.2. of Corollary 5.3, (iv), may be regarded as being essentially a consequence of the — [cf. Remarks 5.1.3, 5.2.2] of the tempered Frobenioid “Kummer-readiness” F v cf. also the arguments applied in the proofs of [AbsTopIII], Proposition 3.2, (iv); [AbsTopIII], Corollary 5.2, (iv). We take this opportunity to rectify a minor oversight in [FrdI]. Remark 5.3.3. “unit-profinite type” The hypothesis that the Frobenioids under consideration be of in [FrdI], Proposition 5.6 — hence also in [FrdI], Corollary 5.7, (iii) — may be removed . Indeed, if, in the notation of the proof of [FrdI], Proposition 5.6, one ′ × Primes = c ∈ · φ p ,where , then one has ), for ∈O c ( A φ writes p p p p ′ 2 ′ ′ ′ c · φ · · φ · c c φ · φ = · c = · φ φ = φ φ · p 2 2 p 2 2 p p p 2 2 p p p · c · φ = c c · c φ = φ · · φ = c · c · · φ · φ p p p 2 2 2 2 p p p 2 2 p p − 1 1 − 2 ′ c · · c —so φ , i.e., c · = c c ,soby · c ,for p ∈ Primes .Thus, φ c = = c 2 p p p 2 p p 2 2 2 def − 1 c ,onemay eliminate the final two paragraphs of the proof of [FrdI], = u taking 2 Proposition 5.6. Let Θ † †  † HT F ) } F =( { , V v ∈ mod v be a Θ -Hodge theater [relative to the given initial Θ-data — cf. Definition 3.6] such † D to the equal is [for simplicity] D -prime- } { -prime-strip D that the associated v v ∈ V † D of the D - ΘNF -Hodge theater in the discussion preceding Example 5.1. strip > Write † F >

147 ̈ INTER-UNIVERSAL TEICHM 147 ULLER THEORY I F -prime-strip for the tautologically associated to this Θ-Hodge theater [cf. the data † } F ” of Definition 3.6; Definition 5.2, (i); Example 3.2, (iii); Example 3.3, “ { V ∈ v v † (i)]. Thus, -prime-strip associated [cf. Remark may be identified with the D D > † . F 5.2.1, (i)] to > Example 5.4. Model - and NF-Bridges. Θ (i) For j J ,let ∈ † † { F F } = j v v ∈ V j j j F -prime-strip whose associated D -prime-strip [cf. Remark 5.2.1, (i)] is equal be an † D , to j † † } = { F F v v ∈ V 〉 J 〈 〈 J 〉 J 〉 〈 J 〉 〈 † an whose associated D -prime-strip we denote by F D -prime-strip [cf. Example 〉 J 〈 5.1, (vii)]. Write ∏ def † † = F F j J J ∈ j ∏ ” is to be understood as denoting the capsule with — where the “formal product † † F .Thus, F may be is given by J index set j for which the datum indexed by j J 〉 〈 † F full poly-isomorphism ,ina natural fashion , via the related to > ∼ † † → F F > 〉 J 〈 † and to via the “diagonal arrow” F J ∏ † † † F F = → F J j 〉 〈 J J ∈ j — i.e., the arrow defined as the collection of data indexed by J for which the ∼ † † F .Thus,we F → datum indexed by is given by the j full poly-isomorphism j 〉 〈 J † † j F F “situated on” the constituent labeled as a copy of of the capsule think of > j † † † F all the “situated in a diagonal fashion on” F as a copy of D ; we think of > J 〉 J 〈 † D . constituents of the capsule J † † ] as being related to F F J [for j ∈ (ii) Note that in addition to thinking of j > ∼ † † † F → F F , we may also regard as being related via the full poly-isomorphism > > j † to F [for j ∈ J ] via the poly-morphism j † † Θ † : → ψ F F > j j † Θ φ [i.e., as discussed in Remark 5.3.1]. Write uniquely determined by j † † Θ † : → F ψ F J >  † Θ for the ψ collection of arrows { } the “lying over” —whichwethinkofas J j ∈ j † Θ † Θ = { } φ φ . collection of arrows J ∈ j  j  †  † †  F F be as in Example 5.1, (iii); δ ). Thus, LabCusp( , D ∈ (iii) Next, let †  ( D )-orbit of [cf. the discussion of Example 4.3, (i)] there exists a unique Aut 

148 148 SHINICHI MOCHIZUKI ∼    † δ  → [  ] ∈ LabCusp( D →D ). We shall refer to isomorphisms that maps D †  D ) [cf. Definition 4.1, (v)] any element that maps to an -valuation ∈ ( as a δ V ± un †  ( [cf. Example 4.3, (i)] via this Aut )-orbit of isomorphisms. D V element of  -valuation may also be defined intrinsically by means Note that the notion of a δ † NF . Indeed, [one verifies immediately that] a φ -NF-bridge of the structure of D  †  maybedefinedasavaluation V ∈ -valuation ( δ ) that lies in the “image” [in D NF † † † D of the unique D of the capsule D -prime-strip φ the evident sense] via j J  ∼  † † † NF LabCusp( such that the bijection LabCusp( D ) ) induced by → φ D [cf. the j  † discussion of Example 4.5, (i)] maps to the element of LabCusp( δ )thatis D j 1 ” , relative to the bijection of the second display of Proposition 4.2. “labeled (iv) We continue to use the notation of (iii). Then let us observe that by †  D ), one may construct, in a natural -valuations δ ∈ at each of the ( localizing V -prime-strip way, an F †  F | δ  † —from — which is well-defined F up to isomorphism [i.e., in the notation of †   × ̃ ( D , equipped with its natural )-action]. Indeed, π O Example 5.1, (iv), from 1 p , this follows by considering the -adic Frobe- v at a nonarchimedean δ -valuation v nioids suitable open subgroups [cf. Remark 3.3.2] associated to the restrictions to ⋂  †   †  † † π )( ⊆ π LabCusp( ( D D ( ) ⊆ π ∈ ( ) [i.e., D D )) determined by δ of Π 1 p 1 1 0 X ” discussed in Definition ”, “ X open subgroups corresponding to the coverings “ − →  † π )- ( is chosen [among its D 3.1, (e), (f); cf. also Remark 3.1.2, (i)], where Π 1 p 0 ,ofthe pairs correspond to v conjugates] so as to  ̃ “Π O ”  p 0 p ̂ 0 of Example 5.1, (v) [cf. also Example 5.1, (vi)]. [Here, we note that, when v bad , one must replace these “ lies over an element of V suitable open subgroups ”by mod their mono-anabelian algorithm implicit tempered analogues , i.e., by applying the in the proof of [SemiAnbd], Theorem 6.6.] On the other hand, at an archimedean δ - , this follows by applying the functorial algorithm for reconstructing the v valuation given in [AbsTopIII], Corollaries 2.8, corresponding Aut -holomorphic orbispace at v ∼ †   †  2.9, together with the discussion concerning the M “isomorphism D M → ( ) ” in Example 5.1, (v) [cf. also Example 5.1, (vi)]. Here, we observe that since ± un fails to be injective → V , in order to relate V the natural projection map mod the restrictions obtained at different elements in a fiber of this map in a well- †  as being well-defined only up to | F defined fashion, it is necessary to regard δ isomorphism. Nevertheless, despite this indeterminacy inherent in the definition †  ‡ | , it still makes sense to define, for an F -prime-strip F F with underlying of δ ‡ poly-morphism D [cf. Remark 5.2.1, (i)], a -prime-strip D †  ‡ F → F ∼ ‡ †   † ∈ F | F for some δ → LabCusp( full poly-isomorphism D to be a ) [cf. Def- δ inition 4.1, (vi)]. Moreover, it makes sense to pre-compose such poly-morphisms F -prime-strips and to post-compose with isomorphisms of such poly-morphisms †  F . Here, we note that such a poly- with isomorphisms between isomorphs of  ‡ † → may be thought of as “lying over” an induced poly-morphism F F morphism  ‡  ‡ † † [cf. Definition 4.1, (vi)], and that any poly-morphism is F → D F → D

149 ̈ INTER-UNIVERSAL TEICHM 149 ULLER THEORY I ‡ by pre-composition with automorphisms of , as well as by post-composition fixed F †  F ). Also, we observe that such a poly-morphism ( Aut with automorphisms ∈   ‡ † → compatible with the local and global F F -coric structures [cf. Ex- κ is ∞ and domain ample 5.1, (v); Definition 5.2, (vi), (viii)] that appear in the codomain of associ- restriction of this poly-morphism in the following sense: the operation of ated [cf. the discussion of Example 5.1, (v); Definition 5.2, (vi), Kummer classes (viii); the constructions discussed in the present item (iv)] determines a collection, ∈ V , of poly-morphisms of pseudo-monoids v indexed by } {  rat  † ‡ ‡ † M D ( M → ) M  ⊆ π κv v × κ κ 1 ∞ ∞ ∞ V v ∈ data in the domain of the arrow that appears in the display is re- —wherethe global garded as only being defined up to automorphisms induced by inner automorphisms † rat  ( D ) [cf. the discussion of Example 5.1, (i)]; the local data in the codomain of π 1 of the arrow that appears in the display is regarded as only being defined up to au- ‡ F [cf. Definition 5.2, automorphisms of the F tomorphisms induced by -prime-strip with respect to equivariant (vi), (viii); Corollary 5.3, (ii)]; the arrow of the display is rat  ‡ rat † D D ) ) [i.e., relative to the respective π ( → ( π the various homomorphisms v 1 1 actions of these groups on the pseudo-monoids in the domain and codomain of the [cf. the constructions discussed in the present item (iv), as well as arrow] induced the theory summarized in [AbsTopIII], Theorem 1.9, and [AbsTopIII], Corollaries arc ‡ †  F → ;when v ∈ V F , we regard the 1.10, 2.8] by the given poly-morphism ‡ ‡ M splittings ⊆ M as being equipped with the various pseudo-monoids κ κv v × ∞ ∞ e } -prime-strips capsule of F is a F discussed in Definition 5.2, (viii). Finally, if { E e ∈ whose associated capsule of D -prime-strips [cf. Remark 5.2.1, (i)] we denote by e } D poly-morphism , then we define a { ∈ E e e † †  e { → { → } } (respectively, F F ) F F ∈ E ∈ e e E e †  e † { to be a collection of poly-morphisms } ) } → (respectively, { F F → F F ∈ E ∈ E e e e  † F F } (respectively, → { [cf. Definition 4.1, (vi)]. Thus, a poly-morphism E e ∈ e † F } ) may be thought of as “lying over” an induced poly-morphism F → { E e ∈ † e e  † D → } (respectively, { D D } → D ) [cf. Definition 4.1, (vi)]. { ∈ e ∈ e E E (v) We continue to use the notation of (iv). Now observe that by Corollary 5.3, (ii), there exists a unique poly-morphism NF †  † † F → : F ψ J  † NF φ . that lies over  (vi) We continue to use the notation of (v). Now observe that it follows from the  † †  definition of in terms of terminal objects F of D [cf. Example 5.1, (iii)] that mod † †  → F F [cf. the notation of (i)] induces, via “restriction” any poly-morphism 〉 J 〈 isomorphism class of functors [cf. Definition 5.2, (i); the [in the evident sense], an notation of Example 5.1, (vii)] ∼     † † † † † → F F F F → ⊇ F ) → ( v mod J 〉 〈 〈 J 〉

150 150 SHINICHI MOCHIZUKI arc for each ∈ V v ,wewrite — where, by abuse of notation when v V ∈ J 〉 〈 〉 〈 〈 〉 〉 J J 〈 J † “ F ” for the category portion of the “collection of data” that appears in Def- v 〈 J 〉 of the choice of the poly-morphism inition 5.2, (i), (b) — which is independent   † † †  F F F → [i.e., among its -conjugates ]. That is to say, the fact that F 〈 〉 J mod l †  terminal objects of [cf. also the definition of F is defined in terms of given D mod in Definition 3.1, (b)!] implies that this particular isomorphism class of functors is immune to that appear in the choice of [i.e., fixed by] the various indeterminacies †  † → F .Letuswrite F 〉 J 〈 ∼   † † †  †  † ( F ⊇ → F F → ) F F → 〈 〉 J mod J 〈 〉 v for the collection of isomorphism classes of restriction functors just defined, as J 〉 〈 ranges over the elements of V . In a similar vein, we also obtain collections of 〈 J 〉 natural isomorphism classes of restriction functors   † † † † F F ; → F F → J j j J for ∈ J . Finally, just as in Example 5.1, (vii), we obtain natural realifications j   R R R  † R R † † † † R † F → F F → → F ; F ; F J j 〈 〉 J j J 〉 〈 J F -prime-strips — i.e., of the various [cf. [FrdI], Corollary 5.4; [FrdII], realifications Theorem 1.2, (i); [FrdII], Theorem 3.6, (i)] of each of the Frobenioid [that is to say, category ] portions of the data of Definition 5.2, (i), (a), (b) — and isomorphism classes of restriction functors discussed so far. the objects constructed in (vi) (vii) We shall refer to as “pivotal distributions” R   † † R † † → F → F F ; F pvt pvt pvt pvt in the case j = 1 — cf. Fig. 5.2 below. v · n · ... ◦ . . . ′ ′ · · v n ... ◦ . . . ′′ ′′ · v · n ... ◦ Fig. 5.2: Pivotal distribution

151 ̈ INTER-UNIVERSAL TEICHM 151 ULLER THEORY I The constructions of Example 5.4, (i), (ii) (respectively, Example Remark 5.4.1. † Θ φ D portion Θ - 5.4, (iii), (iv), (v), (vi), (vii)) manifestly only require the -bridge  D -ΘNF † † NF -NF-bridge D -ΘNF-Hodge theater φ HT portion )ofthe (respectively, D  [cf. Remark 5.1.2]. Remark 5.4.2.  † F and its re- (i) At this point, it is useful to consider the various copies of mod alifications introduced so far from the point of view of “log-volumes” , i.e., arith- [cf., e.g., the discussion of [FrdI], Example 6.3; [FrdI], Theorem 6.4; metic degrees  † may be thought of as F Remark 3.1.5 of the present paper]. That is to say, since j   † † ” F over [cf. ” — i.e., a sort of “section of K over F F a sort of “section of mod J mod the discussion of prime-strips in Remark 4.3.1] — one way to think of log-volumes  †  of — i.e., corresponding, to the F is as quantities that differ by a factor of l 〉 〈 J ∼   † . Put another way, this amounts — from log-volumes of → F F J cardinality of j l  † F as being to thinking of arithmetic degrees that appear in the various factors of J and hence of arithmetic degrees that averaged over the elements of J  † as the . F “resulting averages” appear in 〉 〈 J This sort of averaging may be thought of as a sort of abstract, Frobenioid-theoretic analogue of the normalization of arithmetic degrees that is often used in the theory of heights [cf., e.g., [GenEll], Definition 1.2, (i)] that allows one to work with heights with respect to change in such a way that the height of a point remains invariant of the base field. isomorphisms of Frobenioids (ii) On the other hand, to work with the various ∼   † † — involved amounts [since the F F is an arithmetic degree → —suchas j 〉 〈 J intrinsic invariant of the Frobenioids involved — cf. [FrdI], Theorem 6.4, (iv); Remark 3.1.5 of the present paper] to thinking of arithmetic degrees that appear  † in the various factors of F as being J  [i.e., without dividing by a factor of l summed ] over the elements of J  † and hence of arithmetic degrees that appear in “resulting F as the 〉 J 〈 sums” . This point of view may be thought of as a sort of abstract, Frobenioid-theoretic normalization of arithmetic degrees or heights in which the height analogue of the of a point is multiplied by the degree of the field extension when one executes a change of the base field. The notions defined in the following “Frobenioid-theoretic lifting” of Definition 4.6 will play a central role in the theory of the present series of papers. Definition 5.5. NF-bridge [relative to the given initial Θ-data] to be a collec- (i) We define an tion of data NF ‡ ψ  ‡ ‡   ‡ F F F  −→ ) ( J

152 152 SHINICHI MOCHIZUKI as follows: ‡ ‡ (a) capsule of = } { F , indexed by a finite index is a F F -prime-strips J ∈ j j J ‡ ‡ .Write set J -prime-strips D { } D = D for the associated capsule of J J j j ∈ [cf. Remark 5.2.1, (i)].  ‡ ‡ †  †   (b) F F categories , are F , , respec- F equivalent to the categories ‡   ‡ F , F is equipped with tively, of Example 5.1, (iii). Thus, each of [cf. [FrdI], Corollary 4.11; [FrdI], Theo- natural Frobenioid structure a  ‡  ‡ D for the , D rem 6.4, (i); Remark 3.1.5 of the present paper]; write respective base categories of these Frobenioids. ‡  ‡   morphism (c) The arrow “ ” consists of the datum of a → D D †  § 0] to the natural morphism which is D abstractly equivalent → [cf. †  isomorphism D of Example 5.1, (i), together with the datum of an ∼  ‡ ‡  | F → [cf. the discussion of Example 5.1, (iii)]. F ‡  D NF ‡ (d) is a poly-morphism that lifts [ uniquely ! — cf. Corollary 5.3, (ii)] a ψ   NF ‡ ‡ NF ‡ ‡ poly-morphism → φ such that -NF-bridge φ D : forms a D D J   [cf. Example 5.4, (v); Remark 5.4.1]. Thus, one verifies immediately that any NF-bridge as above determines an as- ‡ NF ‡ ‡  D φ [iso]morphism of D : ). We define a(n) → D ( -NF-bridge sociated J  NF-bridges NF NF 1 2 ψ ψ   2  2 1  1  1 2  → ( −→ F −→  F F  ) F F F ) ( J J 2 1 to be a collection of arrows ∼ ∼ ∼ 2  2 1 2  1   1 ; F F ; F F F → → → F J J 2 1 ∼ 1 2 F → —where F 0], hence induces a § is a capsule-full poly-isomorphism [cf. J J 2 1 ∼ ∼ 2 1 1   2 poly-isomorphism → F D D → poly-isomorphism F ; is a which lifts J J 2 1 ∼ ∼ 2 1 1 2   a poly-isomorphism → D → , D D such that the pair of arrows D J J 1 2 ∼ ∼ 2  1 1   2  -NF-bridges; D F D → → forms a morphism between the associated F D i NF is an [in the evident sense] with the isomorphism ψ compatible — which are [for  i 2], as well as with the respective “  ’s”. It is immediate that any morphism , =1 -NF-bridges. There is D of NF-bridges induces a morphism between the associated an evident notion of composition of morphisms of NF-bridges. -bridge [relative to the given initial Θ-data] to be a collection (ii) We define a Θ of data ‡ Θ ψ  Θ ‡ ‡ ‡ ( F  −→ HT ) F > J as follows: ‡ ‡ (a) is a F F = } , indexed by a finite index -prime-strips F { capsule of J ∈ j j J ‡ ‡ set J .Write = { D D } for the associated capsule of D -prime-strips J j J j ∈ [cf. Remark 5.2.1, (i)]. Θ ‡ HT is a Θ -Hodge theater . (b)

153 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 153 Θ ‡ ‡ [cf. the HT tautologically associated to -prime-strip is the F F (c) > discussion preceding Example 5.4]; we use the notation “  ”todenote Θ ‡ ‡ ‡ this relationship between and F D HT for the D -prime- .Write > > ‡ F [cf. Remark 5.2.1, (i)]. associated to strip > Θ ‡ ‡ Θ ‡ ‡ ‡ Θ (d) } F ψ → is the collection of poly-morphisms ψ ψ = F : { J ∈ j > j  j j ‡ Θ determined [i.e., as discussed in Remark 5.3.1] by a D -Θ-bridge = φ  ‡ Θ ‡ ‡ : . D } → φ D { j J j > ∈ j associated Thus, one verifies immediately that any Θ-bridge as above determines an ‡ Θ ‡ ‡ φ -bridges ). We define a(n) : Θ D D [iso]morphism of → D ( -bridge Θ - J >  Θ Θ 1 2 ψ ψ   Θ Θ 2 2 2 1 1 1 ( HT HT F → )  ) F −→ −→ F F (  J J > > 2 1 to be a collection of arrows ∼ ∼ ∼ Θ Θ 1 1 2 2 1 2 F → F → HT F F ; ; HT → > > J J 2 1 ∼ ∼ 2 1 2 1 F → is a capsule-full poly-isomorphism [cf. § 0]; is F —where F → F > J > J 2 1 ∼ Θ Θ 1 2 full poly-isomorphism the ; HT Θ isomorphism of → -Hodge theaters HT is an i Θ ψ [for compatible [cf. Remark 3.6.2] — which are [in the evident sense] with the  =1 , 2], as well as with the respective “  ’s” [cf. Corollary 5.6, (i), below]. It i is immediate that any morphism of Θ-bridges induces a morphism between the associated -Θ-bridges. There is an evident notion of composition of morphisms D of Θ-bridges. -Hodge theater [relative to the given initial Θ-data] to be (iii) We define a ΘNF a collection of data ‡ Θ NF ‡ ψ ψ   ΘNF Θ ‡ ‡  ‡ ‡ ‡  ‡ F HT ←− F −→  F )  F HT =( > J ‡ ‡   ‡ —wherethedata(  ←− F F F ) forms an NF-bridge ; the data J Θ ‡ ‡ ‡ ( F associated data  — such that the HT −→ )formsaΘ -bridge F > J ‡ Θ NF ‡ -ΘNF-Hodge theater. A(n) , φ φ [iso]morphism of D [cf. (i), (ii)] forms a } {   ΘNF -Hodge theaters is defined to be a pair of morphisms between the respective associated NF- and Θ-bridges that are compatible with one another in the sense that they induce the same bijection between the index sets of the respective capsules of F -prime-strips. There is an evident notion of composition of morphisms of ΘNF- Hodge theaters. Corollary 5.6. (Isomorphisms of Θ - Θ -Hodge Theaters, NF-Bridges, -Hodge Theaters) Relative to a fixed collection of initial Bridges, and ΘNF Θ : -data (i) The natural functorially induced map from the set of isomorphisms be- tween two Θ -Hodge theaters to the set of isomorphisms between the respective associated D -prime-strips [cf. the discussion preceding Example 5.4; Remark 5.2.1, (i)] is . bijective (ii) The natural functorially induced map from the set of isomorphisms be- tween two NF-bridges (respectively, two Θ -bridges ; two ΘNF -Hodge the- aters ) to the set of isomorphisms between the respective associated D -NF-bridges

154 154 SHINICHI MOCHIZUKI - - -bridges ; associated D Θ ΘNF -Hodge theaters )is associated (respectively, D bijective . -bridge, the set of capsule-full poly-isomorphisms Θ (iii) Given an NF-bridge and a between the respective capsules of F -prime-strips which allow one to glue the given  -torsor . -Hodge theater forms an NF- and -bridges together to form a Θ F ΘNF l Proof. First, we consider assertion (i). Sorting through the data listed in the Θ † definition of a Θ-Hodge theater [cf. Definition 3.6], one verifies immediately HT † F [cf. that the only data that is not contained in the -prime-strip associated F > global data of Definition 3.6, (c), and the discussion preceding Example 5.4] is the bad ” [cf. Example 3.2, (i)] at the v ∈ V . F the tempered Frobenioids isomorphic to “ v good v That is to say, for V ∈ , one verifies immediately that † † F F = >,v v [cf. Example 3.3, (i); Example 3.4, (i); Definition 3.6; Definition 5.2, (i)]. On the other hand, one verifies immediately that this global data may be obtained ‡  ‡ → F F  ” summarized in the second by applying the functorial algorithm “ associated F -prime-strips that appear. Thus, display of Remark 5.2.1, (ii), to the assertion (i) follows by applying Corollary 5.3, (ii), to the associated F -prime- bad ∈ V . strips at tempered Frobenioids v and Corollary 5.3, (iv), to the various This completes the proof of assertion (i). In light of assertion (i), assertions (ii), (iii) follow immediately from the definitions and Corollary 5.3, (i), (ii) [cf. also Proposition 4.8, (iii), in the case of assertion (iii)]. © Remark 5.6.1. Observe that the various “functorial dynamics” studied in § 4 — i.e., more precisely, analogues of Propositions 4.8, (i), (ii); 4.9; 4.11 — apply to the NF-bridges ,Θ -bridges ,andΘNF -Hodge theaters studied in the present § 5. Indeed, such analogues follow immediately from Corollaries 5.3, (ii), (iii); 5.6, (ii).

155 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 155 Section 6: Additive Combinatorial Teichm ̈ uller Theory additive analogue — i.e., which revolves § In the present 6, we discuss the labels that arise from around the “functorial dynamics” F ∈ l developed in § 4for —ofthe “multiplicative combinatorial Teichm ̈ uller theory”  . These considerations lead naturally to certain enhancements of the F labels ∈ l § 5. On the other hand, despite the resemblance Hodge theaters various considered in 6tothetheoryof § of the theory of the present § 5, the theory of the present § 4, 6 is, in certain respects — especially those respects that form the analogue of the § 5 — substantially . § theory of technically simpler In the following, we fix a collection of initial Θ -data bad ,l,C V , F/F, X , V ) , ( F mod K as in Definition 3.1; also, we shall use the various notations introduced in Definition 3.1 for various objects associated to this initial Θ-data. Definition 6.1. (i) We shall write def ±  F = F  {± 1 } l l for the group determined by forming the semi-direct product with respect to the ±  × and refer to an element of F that maps to +1 } ↪ → natural inclusion 1 {± F l l  ± (respectively, − 1) via the natural surjection F  {± 1 } as positive (respectively, l ± negative F ). We shall refer to as an -orbit E {± any set } equipped with a -group 1 l ∼ ± of bijections E . Thus, any F F → -group E is equipped with a natural F -module l l l  ± ± structure . We shall refer to as an F any set T equipped with an F -orbit -torsor l l ∼ ±  T of bijections F on F by automorphisms of → F [relative to the action of l l l ± the form F z  →± z + λ ∈ F ]. Thus, if ,for λ ∈ F F -torsor, then the  is an T l l l l given by the translations abelian group of automorphisms of the underlying set of F l F  z  → z + λ ∈ , determines an abelian group ∈ ,for λ F F l l l Aut ( T ) + of “positive automorphisms” of the underlying set of T . Moreover, Aut )is ( T + ± equipped with a natural structure of F -group [such that the abelian group struc- l ) induced by this ( T ) coincides with the F T -module structure of Aut ( ture of Aut + + l ± ± F T is an F -group structure]. Finally, if -torsor, then we shall write l l Aut ( T ) ± for the group of automorphisms of the underlying set of T determined [relative to ± the F -torsor structure on T ] by the group of automorphisms of the underlying set l  ± of F ) is the unique subgroup of index 2]. T given by ( [so Aut Aut ( T ) ⊆ F + ± l l

156 156 SHINICHI MOCHIZUKI (ii) Let † † } { = D D ∈ v v V be a D -prime-strip [relative to the given initial Θ-data]. Observe [cf. the discussion non † ) determines, in a functorial ,then ( V D π ∈ of Definition 4.1, (i)] that if v 1 v good ∈ V ” ] group corresponding to “ X v [in fact, profinite if topological fashion, a v good bad ∈ ; [EtTh], Proposition 2.4, if v ∈ V V ], which contains v [cf. Corollary 1.2 if ± 0 † † ) as an open subgroup; thus, if we write D ( D for B ( − ) of this topological π 1 v v ± † † group, then we obtain a natural morphism [cf. → D D § 0]. In a similar vein, if v v arc v ∈ ,thensince X -core, a routine translation into the “language K admits a V v − → v of Aut-holomorphic orbispaces” of the argument given in the proof of Corollary 1.2 † [cf. also [AbsTopIII], Corollary 2.4] reveals that determines, in a functorial D v ± † fashion, an Aut-holomorphic orbispace X D ”, together with corresponding to “ v v ± † † a natural morphism Aut D → of D -holomorphic orbispaces . Thus, in summary, v v one obtains a collection of data ± ± † † D D = { } v V ∈ v † completely determined by D . (iii) Suppose that we are in the situation of (ii). Then observe [cf. the dis- of cussion of Definition 4.1, (ii)] that by applying the group-theoretic algorithm [AbsTopI], Lemma 4.5 [cf. also Remark 1.2.2, (ii), of the present paper], to the non † ( ) of a cofinal D ( )when v ∈ V − , or by considering π topological group π 0 v 1 collection of “neighborhoods of infinity” [i.e., complements of compact subsets] of arc † D , it makes sense to speak of when v ∈ V the underlying topological space of v ± † † of the set of cusps ; a similar observation applies to v ,for v ∈ V .If D ∈ V , D v v † † D to be the set of cusps of D that lie -label class of cusps of then we define a ± v v over a single cusp [i.e., corresponding to an arbitrary element of the quotient “ Q ” ,l -tors)” given that appears in the definition of a “hyperbolic orbicurve of type (1 ± † D ;write in [EtTh], Definition 2.1] of v ± † LabCusp D ( ) v ± † † ( ) D D . Thus, [for any v ∈ V !] LabCusp -label classes of cusps ± set of of for the v v × F admits a natural action by [cf. [EtTh], Definition 2.1], as well as a zero element l ± † 0 † ∈ η D ) LabCusp ( v v -canonical element and a ± ± † ± † LabCusp ) ( η D ∈ v v well-defined up to multiplication by ± 1, and which may be constructed —whichis † solely from natural bijection [cf. Definition 4.1, (ii)] — such that, relative to the D v { } ∼ ± † † 0 † LabCusp \{ η D D } ) / {± 1 } ) → LabCusp( ( v v v † ± † [cf. the notation of Definition 4.1, (ii)], we have η  → .Inparticular,we η v v obtain a natural bijection ∼ ± † ( D ) → F LabCusp v l

157 ̈ INTER-UNIVERSAL TEICHM 157 ULLER THEORY I well-defined up to multiplication by ± compatible ,relativetothe —whichis 1and )” of the preceding display, with the natural bijec- − natural bijection to “LabCusp( tion of the second display of Proposition 4.2. That is to say, in the terminology of ± ± ± † ( F D -group -group structure .This F ) is equipped with a natural (i), LabCusp v l l natural surjection structure determines a † } D {± 1 )  Aut( v ± † D ( ). Write — i.e., by considering the induced automorphism of LabCusp v † † Aut ( ) ⊆ Aut( D D ) v v + “positive automorphisms” [i.e., the kernel of the above for the index two subgroup of def † † † ) ( =Aut( D ” denotes the ) [i.e., where “ ) \ Aut D ( \ D surjection] and Aut + v v v − ] for the subset of “negative automorphisms” . In a similar set-theoretic complement vein, we shall write † † ( D ) ⊆ Aut( D ) Aut + for the subgroup of [i.e., automorphisms each of whose “positive automorphisms” V V components, for v ], and, if α ∈{± 1 } {± [i.e., where we write ∈ 1 } V ,is positive to {± 1 } ], V for the set of set-theoretic maps from † † Aut D ( ⊆ Aut( ) D ) α “ -signed automorphisms” [i.e., automorphisms each of whose for the subset of α − ∈ V ,is positive if α ( v ) = +1 and negative if α 1]. v )= ( v components, for (iv) Suppose that we are in the situation of (ii). Let ‡ ‡ { = D D } v ∈ V v be another D -prime-strip [relative to the given initial Θ-data]. Then for any v ∈ V , ∼ † ‡ we shall refer to as a + -full poly-isomorphism → any poly-isomorphism D D v v † ‡ of an isomorphism )- [or, equivalently, Aut orbit ( ( D )-] D obtained as the Aut v + v + ∼ † ‡ † ‡ D . In particular, if = D → D D , then there are precisely two +-full poly- v v ∼ ‡ † isomorphisms D D , namely, the +-full poly-isomorphism determined by the → v v positive , and the unique non- identity isomorphism, which we shall refer to as positive +-full poly-isomorphism, which we shall refer to as negative . In a similar ∼ † ‡ any poly-isomorphism → D D -full poly-isomorphism vein, we shall refer to as a + † ‡ of an isomorphism D )- [or, equivalently, Aut orbit ( ( )-] D obtained as the Aut + + ∼ † ‡ † ‡ D . In particular, if D D = → D , then the set of +-full poly-isomorphisms ∼ † ‡ D natural bijective correspondence [cf. the discussion of (iii) above] D → is in ∼ † ‡ V } 1 {± with the set D → D that ; we shall refer to the +-full poly-isomorphism V -signed as the α + -full poly-isomorphism . Finally, a ∈{± α } 1 corresponds to + -full poly-morphism between capsules of D -prime-strips capsule- ∼ ‡ † ′ ′ ′ } →{ D D } { t t t t ∈ ∈ T T is defined to be a poly-morphism between two capsules of -prime-strips determined D ∼ † ‡ t T ∈ ] between the constituent D D [where → -full poly-isomorphisms by + t ( t ) ι ′ objects indexed by corresponding indices, relative to some injection ι : T↪ → T .

158 158 SHINICHI MOCHIZUKI (v) Write def ±  0 X ( = B ) D K finite ́ 0; the situation discussed in Definition 4.1, (v)]. Thus, we have a § etale [cf.    0 ± ( →D [cf. Example C = . Just as in the case of D B ) double covering D K ±  ,the outer D 4.3, (i)], one may construct, in a fashion from category-theoretic homomorphism ±  ) } GL 1 ( F {± ) / → D Aut( l 2 [i.e., from the Galois ac- -torsion points E arising from the of the elliptic curve l F ab tion on Δ F that, relative ]. Moreover, it follows from the construction of X ⊗ l K X ∼  ± Aut( D natural isomorphism to the Aut( X ) [cf., e.g., [AbsTopIII], Theo- ) → K rem 1.9], the image of the above outer homomorphism is equal to a subgroup of [cf. the discussion ( F } ) / {± 1 } that contains a Borel subgroup of SL 1 ( F {± ) / GL l 2 2 l of Example 4.3, (i)] — i.e., the Borel subgroup corresponding to the rank one quo- ab ⊗ F that gives rise to the covering X → X . In particular, this rank tient of Δ K l K X natural surjective homomorphism one quotient determines a   ± D Aut( F )  l ±  D from !] — whose kernel reconstructed category-theoretically [which may be  ±  ± ). One verifies immediately that the sub- ) ⊆ ( D D Aut( we denote by Aut ± ∼  ±  ± group Aut ⊆ Aut( D D ( ) ( → Aut( X ⊆ ) contains the subgroup Aut ) ) X ± K K K . X )of K -linear automorphisms and acts transitively on the cusps of X Aut( K K  ± ±  Aut re- ) ⊆ ) for the subgroup [which may be ( ( D D Next, let us write Aut ± csp  ± ! — cf. [AbsTopI], Lemma 4.5, as well from D constructed category-theoretically fix the cusps as Remark 1.2.2, (ii), of the present paper] of automorphisms that of . Then one obtains natural outer isomorphisms X K ∼ ∼ ±   ±  ± Aut ( Aut ) ) / Aut ( → D ( D ) X → F csp K ± K l [cf. the discussion preceding [EtTh], Definition 2.1] — where the second outer [cf. of C  cusp isomorphism depends, in an essential way, on the choice of the K  ±  ± ) ( Aut ) for the ( D D ⊆ Definition 3.1, (f)]. Put another way, if we write Aut ± +  ± D  determines a ), then the cusp ( unique index two subgroup containing Aut csp ± natural F -group structure on the subgroup l ±  ±   ± ±  Aut ) / ) ) ⊆ Aut ) ( D D ( D / Aut Aut ( D ( + csp ± csp [which corresponds to the subgroup Gal( X /X natural ) ⊆ Aut ) via the ( X K K K K outer isomorphisms of the preceding display] and, in the notation of (vi) below, a ± ± ±  on the set LabCusp ( ). Write D -torsor structure F natural l def ±  ±  ± V =Aut ) ) · V =Aut K ( D ( D ) · V ⊆ V ( csp ± [cf. the discussion of Example 4.3, (i); Remark 6.1.1 below] — where the “=” follows immediately from the natural outer isomorphisms discussed above. Then [by ⋂ bad ± V V considering what happens at the elements of ] one verifies immediately  ±  ± ∼ ) may be identified with ) X Aut( D ( D ) ⊆ Aut( that the subgroup Aut = ± K ± V . )that stabilizes X the subgroup of Aut( K

159 ̈ INTER-UNIVERSAL TEICHM 159 ULLER THEORY I (vi) Let † ±  D  ± D be any category isomorphic to . Then just as in the discussion of (iii) in the non V ”, it makes sense [cf. [AbsTopI], Lemma 4.5, as well as Remark ∈ case of “ v †  ± D ,aswellasthe 1.2.2, (ii), of the present paper] to speak of the set of cusps of set of -label classes of cusps ± ± †  ± ( ) D LabCusp  † ± . D — which, in this case, may be identified with the set of cusps of Θ -approach” dis- (vii) Recall from [AbsTopIII], Theorem 1.9 [applied via the “  — cf. the discussion of cussed in Remark 3.1.2], that [just as in the case of D Definition 4.1, (v)] there exists a group-theoretic algorithm for reconstructing, from ±  set D )[cf. § ( F ” of the base field “ K ”, hence also the 0], the algebraic closure “ π 1  ± of valuations “ V ( F F [e.g., as a collection of topologies on )” from — cf., e.g., D arc V ( K ) , let us recall [cf. Remark ∈ w [AbsTopIII], Corollary 2.8]. Moreover, for reconstruct group-theoretically , 3.1.2; [AbsTopIII], Corollaries 2.8, 2.9] that one may  ± ±  † .Let -holomorphic orbispace X be associated to X D ( ), the Aut D from π 1 w w as in (vi). Then let us write †  ± V ( ) D †  ± V for the set of valuations [i.e., “ ( ( )”], equipped with its natural D π F )-action, 1 def † ±   †  ± ± † D ) D = ) /π V ( ( D ) ( V 1 arc ±  † †  ± ±  † for the quotient of V )by ) [i.e., “ V ( K )”], and, for w ∈ π ( D D ( V ) ( , D 1  ± † ( ,w ) D X ” — cf. the discussion of [AbsTopIII], Definition 5.1, (ii)] for the Aut- X [i.e., “ w holomorphic orbispace obtained by applying these group-theoretic reconstruction ±  † π algorithms to ( ). Now if U is an arbitrary Aut -holomorphic orbispace ,then D 1 morphism let us define a †  ± D → U to be a morphism of Aut-holomorphic orbispaces [cf. [AbsTopIII], Definition 2.1, †  ± †  ± arc ∈ D V ( ( D w ,w ) )forsome . Thus, it makes sense to speak of (ii)] U X → ±  † D → the pre-composite (respectively, post-composite) of such a morphism U with a morphism of Aut-holomorphic orbispaces (respectively, with an isomorphism ∼ ‡ †   ± ± ± ‡   ± 0] [cf. § ]). D is a category equivalent to D D D → [i.e., where Remark 6.1.1. In fact, in the notation of Example 4.3, (i); Definition 6.1, (v), it ± ± un = V )). K ( ⊆ V ( V is not difficult to verify [cf. Remark 3.1.2, (i)] that ± -Bridges. Example 6.2. Model Base- Θ ± as an -group [relative to the tautological F F (i) In the following, let us think of l l ± F -group structure]. Let l } = {D {D ; D = } D V  ,v t V v v ∈ ∈ v  t

160 160 SHINICHI MOCHIZUKI ∈ —where v F to denote the pair ( t, v ) [cf. Example t , and we use the notation l t } [cf. Examples -prime-strip” {D 4.3, (iv)] — be copies of the “tautological D ∈ v v V ,write F ∈ t 4.3, (iv); 4.4, (ii)]. For each l ± ± Θ Θ D D D →D : → ; φ : φ ,v  t v  v t t t for the respective positive + -full poly-isomorphisms , i.e., relative to the respective for the D .Write } -prime-strip” {D D identifications with the “tautological ± V v v ∈ } [cf. the constructions of Example 4.4, (iv)] and { D capsule F t t ∈ l ± Θ φ D : → D  ± ± ± Θ φ for the collection of poly-morphisms { } . ∈ F t t l (ii) The collection of data ± Θ ) ,φ , D D (  ± ± − natural poly-automorphism of order two admits a 1 defined as follows: the F l − acts on F as multiplication by 1 and induces the poly- 1 poly-automorphism − F l l ∼ ∼ isomorphisms D D determined [i.e., relative to D [for t ∈ F → ]and D →  t  − l t ]by } the respective identifications with the “tautological -prime-strip” D {D V ∈ v v ∈ V is negative . One verifies the + -full poly-automorphism whose sign at every v [in the evident sense] with compatible , defined in this way, is immediately that − 1 F l ± Θ . φ ± ± V Θ natural poly-automorphism .Then α determines a α of 1 α } ∈{± (iii) Let 1 order 2 } of the collection of data ∈{ , ± Θ D ,φ , ) ( D  ± ± ± Θ acts on F as the identity and on D ,for as follows: the poly-automorphism α t l . One verifies immediately -full poly-automorphism + ,and D -signed as the α F ∈ t  l ± ± Θ Θ compatible . φ [in the evident sense] with , defined in this way, is that α ± ell Example 6.3. Model Base- Θ -Bridges. ± F (i) In the following, let us think of F as an -torsor [relative to the tauto- l l ± logical F -torsor structure]. Let l } {D = D v ∈ V t v t  ± F [for ]and D as in Definition 6.1, (v). In the be as in Example 6.2, (i); D t ∈ ± l ± fix an isomorphism of following, let us F -torsors l ∼ ± ±  LabCusp ( D → F ) l ±  ± [cf. the discussion of Definition 6.1, (v)], which we shall use to LabCusp ( D ) identify isomorphism of groups . Note that this identification induces an F with l ∼ ±   ±  ± Aut ( D ) / ) Aut → F ( D csp ± l

161 ̈ INTER-UNIVERSAL TEICHM 161 ULLER THEORY I [cf. the discussion of Definition 6.1, (v)], which we shall use to identify the group ⋂ good non ±   ±  ± V ( F / Aut (respectively, .If v ∈ V ( D D ) ) with the group Aut ± csp l bad ∈ V ), then the natural restriction functor on finite ́ etale coverings arising from v → X → → X X → X (respectively, X the natural composite morphism v K v − → v v ell Θ φ natural morphism ) determines [cf. Examples 3.2, (i); 3.3, (i)] a → : D X v ,v • K arc  ± ∈ V , then [cf. Example 3.4, (i)] [cf. the discussion of Example 4.3, (ii)]. If v D ∼ ±  tautological morphism we have a D ,v → X = X → X ( D ), hence a morphism v v → − v ell Θ  ± : D ∈ →D , V [cf. the discussion of Example 4.3, (iii)]. For arbitrary v φ v ,v • write ell Θ ±  φ →D : D v v 0 0  ± poly-morphism →D given by the collection of morphisms D of the form for the v 0 ell Θ ◦ φ β ◦ α • ,v  ± D ); we apply the tautological identification ); ( ∈ Aut ( D β α Aut ∈ —where csp v + 0 with D [cf. the discussion of Example 4.3, (ii), (iii), (iv)]. Write of D v v 0 ell Θ ±  φ →D D : 0 0 ell Θ  ± : for the [cf. φ D →D determined by the collection poly-morphism } { V ∈ v v v 0 0 the discussion of Example 4.3, (iv)]. Note that the presence of “ β ” in the defini- ell ell Θ Θ post-compose φ with an element of implies that it makes sense to φ tion of v 0 0 ∼   ± ± ±  ±  ( ( D ,letuswrite D ) ) → F / Aut F .Thus,forany t ∈ F ⊆ Aut l ± csp l l ell Θ ±  φ →D : D t t ell Θ post-composing with the for the result of [i.e., action via poly- φ “poly-action” 0  ± [and pre-composing with the tautological identification t on automorphisms] of D with D ]and D of 0 t ell ± Θ  φ D →D : ± ± ell Θ for the collection of arrows { φ } . ∈ t F t l  ± γ natural poly-automorphism D of .Then γ determines a ∈ γ (ii) Let F ± ± l  ± acts on F F via the usual action of as follows: the automorphism γ and, on F l l ± l ∼ for t ∈ F , induces the + -full poly-isomorphism D sign whose → D at every l t ( t ) γ V γ [cf. the construction of Example 6.2, (ii)]. Thus, we ∈ is equal to the sign of v ±  poly-action F obtain a natural of D . On the other hand, the isomorphism on ± l ∼ ±  ±   ± ±  Aut of (i) determines a natural ( Aut ) / → F ) D ( D poly-action of F ± csp l l ell  ± Θ . Moreover, one verifies immediately that D φ on with respect is equivariant ±  ± ±  natural on D and D ; in particular, we obtain a F to these poly-actions of ± l poly-action ell  ± Θ  ± D ) , D  ( ,φ F ± ± l ell  ± Θ  ± ,φ ) [cf. the discussion of Example D on the collection of data ( D , of F ± ± l 4.3, (iv)].

162 162 SHINICHI MOCHIZUKI def  ± In the following, we shall write l l +1=( l +1) / 2. [Here, Definition 6.4. =  ” was introduced at the beginning of § 4.] l we recall that the notation “ ± ± Θ (i) We define a base- -bridge -bridge D - Θ ,or , [relative to the given initial Θ-data] to be a poly-morphism ± † Θ φ ± † † D −→ D  T ± † † † -prime-strip ; T is an F D —where -group ; is a D capsule = { D D is a } t T t T  ∈ l -prime-strips — such that there exist of T D , indexed by [the underlying set of] isomorphisms ∼ ∼ † † D , D → D → D ±  T  ∼ induced by the second → — where we require that the bijection of index sets F T l ± -groups — conjugation by which maps isomorphism of F isomorphism determine an l ± ± Θ † Θ φ . In this situation, we shall write  → φ ± ± † D | | T ± † l -capsule for the D -capsule obtained from the by forming the quotient | T | of l T the index set T of this underlying capsule by the action of {± 1 } and identifying † D indexed by the elements in the fibers of the the components of the capsule T ± ± Θ † † Θ | T  | via the constituent poly-morphisms of T quotient φ φ [so = { } t T ∈ t ± † is only well-defined up to a positive auto- D each constituent -prime-strip of D | | T , but this will not affect applications of this construction morphism indeterminacy — cf. Propositions 6.7; 6.8, (ii); 6.9, (i), below]. Also, we shall write †  D T def   -capsule determined by the subset T for the l = | T |\{ 0 } of nonzero elements of ± -bridges | [iso]morphism of D - Θ T . We define a(n) | ± ± † Θ Θ ‡ φ φ ± ± ‡ † † ‡ ′ −→ → ) D −→ ( D D ) D ( T T   to be a pair of poly-morphisms ∼ ∼ ‡ ‡ † † ′ ; → D D D D →  T  T ∼ † ‡ ′ capsule- D D → is a —where + -full poly-isomorphism whose induced mor- T T ∼ ∼ ± † ‡ ′ isomorphism of → is an is a -groups ; T D D F → phism on index sets T   l ± ± ‡ † Θ Θ φ , φ .Thereisan with -full poly-isomorphism + compatible — which are ± ± ± -bridges. evident notion of composition of morphisms of D -Θ ell ell Θ (ii) We define a base- “base- Θ -elliptic-bridge” ], or D - Θ -bridge - [i.e., a , [relative to the given initial Θ-data] to be a poly-morphism bridge ell † Θ φ ± † †  ± −→ D D T ± † †  ± † ±  D category equivalent to T is an F is a } -torsor ; D D ; = { —where D t t ∈ T T l is a capsule of D -prime-strips , indexed by [the underlying set of] T — such that there exist isomorphisms ∼ ∼ ± ± † †   D , D D → → D ± T

163 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 163 ∼ → induced by the second — where we require that the bijection of index sets T F l ± -torsors — conjugation by which F isomorphism of isomorphism determine an l ell ell † Θ Θ ell maps φ φ . We define a(n) [iso]morphism of D - Θ → -bridges  ± ± ell ell Θ † Θ ‡ φ φ ± ±  † ±  ‡ ‡ † ± ′ −→ D ) −→ D D ( → ) D ( T T to be a pair of poly-morphisms ∼ ∼  † ‡  ± † ‡ ± ′ → ; → D D D D T T ∼ ‡ † ′ -full poly-isomorphism D D → is a capsule- + —where whose induced mor- T T ∼ ± ′  †  ± ‡ ± T -torsors ; is a D is an isomorphism of → F D → T phism on index sets l  ± † ‡  ± )-] )- [or, equivalently, Aut D ( ( D orbit poly-morphism which is an Aut csp csp ell ell † Θ ‡ Θ , φ φ . There is an evident compatible of isomorphisms — which are with ± ± ell -bridges. notion of composition of morphisms of D -Θ ell ± ± ell (iii) We define a Θ base- Θ -Hodge theater - -Hodge theater ,[relative ,or D to the given initial Θ-data] to be a collection of data ell ± † † Θ Θ φ φ ± ell ± ± D -Θ † † † †  ± ←− D D −→ ) D HT =( T  ± ell ± † Θ ell ± † Θ F is an —where T is a D -Θ φ -bridge; -group φ ; -bridge [relative -Θ is a D ± ± l ± ± to the F F -group structure on T ]—suchthat -torsor structure determined by the l l there exist isomorphisms ∼ ∼ ∼ † ±  † †  ± ; D → D D D D → → ; D T  ±  ell ± ± ell Θ Θ Θ † † Θ   φ φ → , φ φ conjugation by which maps .A(n) [iso]morphism of → ± ± ± ± ± ell is defined to be a pair of morphisms between the respective -Hodge theaters - D Θ ± ell with one another in the -and D -Θ -bridges that are compatible D associated -Θ same poly-isomorphism sense that they induce the between the respective capsules D -prime-strips. There is an evident notion of composition of morphisms of D - of ± ell -Hodge theaters. Θ The following additive analogue of Proposition 4.7 follows immediately from the various definitions involved. Put another way, the content of Proposition 6.5 below may be thought of as a sort of of the constructions carried “intrinsic version” out in Examples 6.2, 6.3. (Transport of ± -Label Classes of Cusps via Base- Proposition 6.5. Let Bridges) ell ± † Θ Θ † φ φ ell ± ± ± -Θ D  † † † ± † ←− D ) −→ D D HT =( T  ± ell [relative to the given initial -Hodge theater -data]. Then: Θ Θ - be a D ell ell † Θ v (i) For each T ,the D - ∈ V -bridge , φ t ∈ induces a [single, well- Θ ± defined!] bijection of sets of ± -label classes of cusps ell ∼ ± ± † Θ ±  † † D D ζ ) ) → LabCusp ( ( : LabCusp v v t t

164 164 SHINICHI MOCHIZUKI ± compatible that is with the respective -torsor structures. Moreover, for w ∈ V , F l the bijection ell ell ell ∼ def ± ± † † Θ † † † Θ − 1 Θ D ( ζ ) ) D ( ) : LabCusp ζ ( =( ◦ LabCusp ) ξ → v w ,w w v v t t t t t t ± F is compatible with the respective structures. Write -group l ± † ( D ) LabCusp t ± ± ± † F for the -group obtained by identifying the various ( F D -groups LabCusp ) , v l l t ell † Θ v as V , via the various ranges over the elements of ξ . Finally, the various ,w v t t ell † Θ determine a [single, well-defined!] ζ bijection v t ell ∼ ± ± † Θ † †  ± ( → LabCusp : LabCusp ( D D ζ ) ) t t ± -torsor structures. — which is compatible with the respective F l ± Θ † ± (ii) For each v V ,the - T ∈ -bridge t φ , D ∈ induces a [single, well- Θ ± defined!] bijection of sets of ± -label classes of cusps ± ∼ ± ± Θ † † † ) D D : LabCusp ) ζ → LabCusp ( ( ,v v  v t t ± that is with the respective F compatible -group structures. Moreover, for w ∈ V , l the bijections ± ± ell ± def ∼ ± ± † † † Θ † Θ † † − 1 Θ Θ ζ LabCusp → D ) ◦ ( ( ζ ( ξ D ) ) ◦ =( : LabCusp ξ ); ,v  ,w  ,w ,w  v w v ,v 0 0 0 0 ± ± ± ± def ∼ ± ± − 1 † Θ † † † Θ Θ † Θ † ξ D ξ LabCusp → ( ◦ ( ) ζ ) =( ) D ) : LabCusp ◦ ( ζ w v ,w ,v  v w v ,w t t t t t t ± -group T 0 ” for the zero element of the F — where, by abuse of notation, we write “ l ± ± † Θ compatible with the respective F —are ξ structures, and we have -group = v ,w l t t ell † Θ . Write ξ ,w v t t ± † LabCusp ( D )  ± ± ± † -group obtained by identifying the various -groups LabCusp ( D F ) , F for the  ,v l l ± † Θ as v ξ ranges over the elements of V , via the various , T . Finally, for any t ∈ ,w  ,v ± ell † Θ † Θ , ζ ζ bijection determine, respectively, a [single, well-defined!] the various v v t t ± ∼ ± ± † Θ † † ( ) : LabCusp → LabCusp D ( ζ D ) t  t ± -group structures. — which is compatible with the respective F l (iii) The assignment ell ± † ±  † Θ →  t  T ζ ( (0) D ∈ LabCusp ) t ± -group 0 ” for the zero element of the F — where, by abuse of notation, we write “ l ± † LabCusp D ( bijection ) — determines a [single, well-defined!] t ∼ ± † ± − 1 †  ( ) ζ LabCusp ) ( : D T → ±

165 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 165 † ζ ]—whichis [i.e., whose inverse we denote by compatible with the respective ± ± F , the composite bijection ∈ t -torsor structures. Moreover, for any T l ell ± ± ell ∼ ± ± − Θ 1 − 1 † Θ † † Θ † † † Θ ◦ ( ◦ ζ ◦ ( ) ) : LabCusp ζ ( ( D ) ) ) → LabCusp ζ ( ) D ζ ( 0 0 0 t 0 t ± † coincides with the automorphism of the set LabCusp D ) determined, relative to ( 0 ell ± † Θ − 1 † − 1 F the . -group structure on this set, by the action of ζ ζ )) ) ) ( ( t (( ± 0 l ell † †  ±  ± † Θ ( ( D D ) . Then if one replaces / φ Aut ) ◦ by α α (iv) Let Aut ∈ csp ± ± ell ell † Θ † Θ ζ [cf. Proposition 6.6, (iv), below], then the resulting “ φ ” is related to the t ± ell ell Θ † † Θ “ ζ ” determined by the original α by post-composition with the image of φ ± t natural bijection [cf. the discussion of Definition 6.1, (v)] via the ∼ ± ±  ± †   † ± †  ± ∼ ( ) ) / → Aut F (LabCusp ) ( D D ( Aut )) ( D Aut = ± csp ± l † ±  ±  † determined by the tautological action of Aut Aut ) ( / D ) D ( on the set of ± csp ±  ± † -label classes of cusps LabCusp ± D ( ) . Next, let us observe that it follows immediately from the various definitions involved [cf. the discussion of Definition 6.1; Examples 6.2, 6.3], together with the poly-automorphisms explicit description of the various discussed in Examples 6.2, (ii), (iii); 6.3, (ii) [cf. also the various properties discussed in Proposition 6.5], that additive we have the following analogue of Proposition 4.8. ± ell -Bridges, Base- Θ -Bridges, Proposition 6.6. (First Properties of Base- Θ ± ell initial -Hodge Theaters) Relative to a fixed collection of - Θ Θ and Base- : data ± -bridges forms a torsor (i) The Θ set of isomorphisms between two D - over the group ) ( V {± 1 } 1 {± }× — where the first (respectively, second) factor corresponds to poly-automorphisms of the sort described in Example 6.2, (ii) (respectively, Example 6.2, (iii)). Moreover, the first factor may be thought of as corresponding to the induced isomorphisms of ± -groups between the index sets of the capsules involved. F l ±  ell D (ii) The Θ set of isomorphisms between two - F forms an -bridges - l torsor — i.e., more precisely, a torsor over a finite group that is equipped with a ±  . Moreover, this set of isomorphisms maps F natural outer isomorphism to l bijectively , by considering the induced bijections, to the set of isomorphisms of ± -torsors between the index sets of the capsules involved. F l ell ± between D - Θ set of isomorphisms (iii) The two -Hodge theaters forms a {± 1 } -torsor . Moreover, this set of isomorphisms maps bijectively , by consid- ± ering the induced bijections, to the set of isomorphisms of F -groups between the l index sets of the capsules involved.

166 166 SHINICHI MOCHIZUKI ell ± + D - Θ - -bridge, the set of capsule- -bridge and a -full poly- Θ (iv) Given a D -prime-strips which allow one to D isomorphisms between the respective capsules of ell ± ell ± -and -Hodge theater - Θ D -bridges together to form a D - Θ - glue the given Θ D forms a torsor over the group ) ( ±  V 1 } {± × F l  ± }× of (ii); the subgroup {± 1 F — where the first factor corresponds to the l ( ) V {± 1 } corresponds to the group of (i). Moreover, the first factor may be thought ± F of as corresponding to the induced isomorphisms of between the index -torsors l sets of the capsules involved. ell (v) Given a D - Θ -bridge, there exists a [relatively simple — cf. the discus-  ± for constructing, up to an F sion of Example 6.2, (i)] functorial algorithm - l ell ± ell -bridge a D - Θ -Hodge Θ D [cf. (ii), (iv)], from the given indeterminacy - ell ell -bridge is the given D - Θ -bridge. whose underlying D - theater Θ   − ] < − 1 < 0 < 1 < 2 <...

167 ̈ INTER-UNIVERSAL TEICHM 167 ULLER THEORY I ± ell aΘ -Hodge theater — is illustrated in Fig. 6.1 above. Thus, Fig. 6.1 may be thought of as a sort of analogue of the additive multiplicative situation illustrated ± , while -bridge ⇑ ” corresponds to the associated [ D -]Θ in Fig. 4.4. In Fig. 6.1, the “ ell ± ’s” denote -prime- -bridge ;the“ / D -]Θ ⇓ the “ D ” corresponds to the associated [ strips. ± -Bridges) Rel- Θ -Bridges Associated to Base- Θ Proposition 6.7. (Base- -data ,let ative to a fixed collection of initial Θ ± † Θ φ ± † † −→ D D  T † † ±  -bridge , as in Definition 6.4, (i). Then by replacing be a by - D Θ D [cf. D T T † † D D with the D -prime-strip -prime-strip D Definition 6.4, (i)], identifying the 0  ± † Θ † [cf. the discussion of Definition 6.4, (i)] to form a D -prime-strip φ D via , > 0 ± good † Θ replacing the various + -full poly-morphisms that occur in at the v ∈ V φ ± by the corresponding full poly-morphisms, and replacing the various + -full poly- ± bad † Θ morphisms that occur in ∈ φ V v by the poly-morphisms described [via at the ± !] in Example 4.4, (i), (ii), we obtain a functorial group-theoretic algorithms for constructing a [well-defined, up to a unique isomorphism!] - Θ - D algorithm bridge † Θ φ  † †  −→ D D > T D -bridge is related to the as in Definition 4.6, (ii). Thus, the newly constructed - Θ ± -bridge via the following correspondences: Θ - D given † † † † †  D D D  → | D D → ;  , 0 > T  ) T ( 0 } \{ T precisely two D -prime-strips to a single — each of which maps -prime-strip. D The various assertions of Proposition 6.7 follow immediately from the Proof. © various definitions involved. additive analogues of Propositions 4.9, 4.11; Corollary 4.12. Next, we consider Proposition 6.8. (Symmetries arising from Forgetful Functors) Relative to a fixed collection of initial Θ -data : ell ± ell Θ (Base- (i) Θ -Hodge the- - D -Bridges) The operation of associating to a ell ell ± -Hodge theater determines a -bridge of the D - Θ nat- D Θ ater the underlying - ural functor category of category of ell ± ell -Hodge theaters -Θ D D -Θ -bridges → and isomorphisms of and isomorphisms of ell ell ± -Θ -Hodge theaters D -Θ D -bridges ell † Θ φ ± ell ± -Θ D † † †  ± ( ) D −→ HT D  → T

168 168 SHINICHI MOCHIZUKI  ± -symmetry — i.e., more precisely, a symme- whose output data admits an F l natural outer try given by the action of a finite group that is equipped with a  ± — which acts doubly transitively [i.e., transitively with F to isomorphism l stabilizers of order two] on the index set [i.e., “ T ”] of the underlying capsule of † D ”] of this output data. -prime-strips [i.e., “ D T ± ell - Θ (Holomorphic Capsules) The operation of associating to a D (ii) - ± ell D -Θ † ± Hodge theater l -capsule the HT † D | | T ± ell D -Θ ± † HT - Θ associated to the underlying -bridge of [cf. Definition 6.4, (i)] D natural functor determines a ± l category of category of -capsules ell ± -Hodge theaters D D of -Θ -prime-strips → and capsule-full poly- and isomorphisms of ± ± ell -Hodge theaters -Θ isomorphisms of l -capsules D ell ± -Θ D † † → D HT  T | | ± ± for the symmet- whose output data admits an -symmetry [where we write S S l l ± | transitively on the index set [i.e., “ | T letters] which acts ”] of this ric group on l output data. Thus, this functor may be thought of as an operation that consists of ∼ →| } | T | = F | / {± 1 | F [i.e., forgetting the bijection F ∈| labels the forgetting l l l ± F determined by the -group structure of T — cf. Definition 6.4, (i)]. In particular, l † D up to isomorphism, then there is a total if one is only given this output data | T | ± T to which a given index possibilities for the element ∈| F |∈| | | | t l of precisely l corresponds, prior to the application of this functor. (Mono-analytic Capsules) By composing the functor of (ii) with the (iii) operation discussed in Definition 4.1, (iv), one obtains a mono-analyticization natural functor ± category of -capsules l category of  ell ± D of D -Hodge theaters -prime-strips -Θ → and capsule-full poly- and isomorphisms of ± ell ± -Θ D isomorphisms of l -capsules -Hodge theaters ell ± D -Θ † †  HT  → D | T | whose output data satisfies the same symmetry properties with respect to labels as the output data of the functor of (ii). Proof. Assertions (i), (ii), (iii) follow immediately from the definitions [cf. also Proposition 6.6, (ii), in the case of assertion (i)]. ©

169 ̈ INTER-UNIVERSAL TEICHM 169 ULLER THEORY I ± ± ± ± ± ± ± ± ± ± ... / / / ↪ → ... ↪ → / / / → / ↪ / / ↪ → / ± -procession of D -prime-strips l Fig. 6.2: An Proposition 6.9. (Processions of Base-Prime-Strips) Relative to a fixed Θ -data : collection of initial ± † Θ ± † † → : φ D -bridge D , Θ (Holomorphic Processions) D (i) Given a -  T ± † D [cf. Definition 6.4, (i)], denote D -prime-strips with underlying capsule of T † ± D [cf. Fig. 6.2, where each ) -prime-strips l the -procession of D Prc( by T ± D -prime-strip] determined by considering the [“sub”]capsules of ” denotes a / “ ± † the capsule ⊆ of Definition 6.4, (i), corresponding to the subsets S D ... ⊆ | T | 1 def ± ± = { 0 , 1 , 2 ,...,t − 1 }⊆ | ⊆ S [where, by abuse of notation, we use the F | = ... S ± l t l notation for nonnegative integers to denote the images of these nonnegative integers ∼ ± F | ], relative to the bijection | -group structure of | T →| F determined by the | in | F l l l ± † † Θ [cf. Definition 6.4, (i)]. Then the assignment T  → Prc( φ D determines a ) T ± natural functor category of category of processions ± D -bridges -prime-strips D of -Θ → and morphisms of and isomorphisms of ± -Θ D processions -bridges ± Θ † †  Prc( D ) φ → T ± ± ∈{ 1 ,...,l whose output data satisfies the following property: for each n ,there } | to which a given index of for the element ∈| F are precisely n possibilities l the index set of the -capsule that appears in the procession constituted by this n output data corresponds, prior to the application of this functor. That is to say, , of cardinalities of “sets of possibilies”, | by taking the product, over elements of F | l one concludes that by considering processions — i.e., the functor discussed above, possibly ell ± ± -Θ D † Θ †  HT φ → that associates to a pre-composed with the functor ± ± ± ell -Hodge theater its associated -bridge — the indeterminacy D - Θ Θ D - ± ( ± l ) l ( consisting of ) possibilities that arises in Proposition 6.8, (ii), is ± possibilities ! . l consisting of a total of to an reduced indeterminacy (Mono-analytic Processions) (ii) By composing the functor of (i) with the mono-analyticization operation discussed in Definition 4.1, (iv), one obtains a natural functor category of processions category of  ± -Θ -bridges -prime-strips D of D → and isomorphisms of and morphisms of ± -bridges -Θ D processions ± †  Θ † Prc( → φ  D ) ± T

170 170 SHINICHI MOCHIZUKI whose output data satisfies the same indeterminacy properties with respect to labels as the output data of the functor of (i). , respectively, with the functors of (iii) The functors of (i), (ii) are compatible Proposition 4.11, (i), (ii), relative to the functor [i.e., determined by the functorial algorithm] of Proposition 6.7, in the sense that the natural inclusions  ± { 1 ,...,j } ↪ → S 1 1 = { 0 , } = − ,...,t S t j def  +1 t } and — j = ∈{ ,...,l 1 [cf. the notation of Proposition 4.11] — where j determine natural transformations ) ( ± Θ † † †   D → ) ) ↪ → Prc( Prc( D φ T T ± ( ) ± †  Θ  † † D  → φ ) ↪ → Prc( Prc( D )  ± T T from the respective composites of the functors of Proposition 4.11, (i), (ii), with the functor [determined by the functorial algorithm] of Proposition 6.7 to the functors of (i), (ii). Assertions (i), (ii), (iii) follow immediately from the definitions. © Proof. The following result is an immediate consequence of our discussion. ± ell ́ -Hodge Theaters) Relative Θ Corollary 6.10. ( Etale-pictures of Base- initial -data to a fixed collection of : Θ functor (i) Consider the [composite] ± ell D -Θ  † † † D  → → D HT  > > ell ± ell ± Θ -Hodge theaters and isomorphisms of - Θ D — from the category of -Hodge - D  -prime-strips and isomor- theaters [cf. Definition 6.4, (iii)] to the category of D  ell ± D Θ -prime-strips — obtained by assigning to the - -Hodge theater D phisms of ± ell D -Θ  † † [cf. Definition 4.1, (iv)] mono-analyticization D of the D - HT the > † D associated, via the functorial algorithm of Proposition 6.7, to the prime-strip > ± ell ell ± ± ell D -Θ -Θ D D -Θ ell † ‡ ± ± † Θ - D underlying HT HT of Θ HT are D - .If , -bridge - ± ell ± ell - -, or D Θ -, link Hodge theaters Θ , then we define the base- ell ± ell ± D -Θ D -Θ D ‡ † −→ HT HT ± ell ell ± -Θ -Θ D D ‡ † HT HT to be the full poly-isomorphism from to ∼  † ‡  D → D > >  D between the -prime-strips obtained by applying the functor discussed above to ± ell ell ± D -Θ -Θ D ‡ † HT HT , .

171 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 171 (ii) If ± ell ell ± ± ell D D D D D -Θ -Θ D D -Θ ( n +1) 1) − n ( n ... HT −→ −→ HT −→ ... HT −→ ell ± ± ell -linked D - Θ -Hodge theaters n ]isan infinite chain of D - Θ ∈ [where Z chain of [cf. the situation discussed in Corollary 3.8], then we obtain a resulting full poly-isomorphisms ∼ ∼ ∼ n   ( n +1) D → → D → ... ... > >  -prime-strips ob- [cf. the situation discussed in Remark 3.8.1, (ii)] between the D tained by applying the functor of (i). That is to say, the output data of the functor of (i) forms a constant invariant [cf. the discussion of Remark 3.8.1, (ii)] — mono-analytic core [cf. the situation discussed in Remark 3.9.1] — of the i.e., a above infinite chain. ± ± ± / / → ↪ → ... / ↪ ... ... | — — ± ±  ±  ± ± ± ... ... ↪ > → = { 0 , / / → / / "} ↪ → / ↪ / → ↪ | ... ... ± ± ± / ↪ ↪ → / → / ... ± ell ́ Fig. 6.3: Etale-picture of -Hodge theaters -Θ D ± ell D - Θ (iii) If we regard each of the -Hodge theaters of the chain of (ii) as spoke a emanating from the mono-analytic core discussed in (ii), then we ob- ell ± —asin -Hodge theaters D Θ etale-picture of ́ — i.e., an diagram tain a -  ” > Fig. 6.3 [cf. the situation discussed in Corollary 3.9, (i)]. In Fig. 6.3, “ denotes the mono-analytic core, obtained [cf. (i); Proposition 6.7] by identifying ± ell -Hodge theater labeled “ 0 ”and Θ the mono-analyticized D -prime-strips of the D - ± ± ± → / ... / ↪ ↪ → ” denotes the “holomorphic” processions of Proposition “ ”; “ / " D - 6.9, (i), together with the remaining [“holomorphic”] data of the corresponding ± ell -Hodge theater. In particular, the mono-analyticizations of the zero-labeled Θ ± ” in the pro- D -prime-strips corresponding to the first “ / D -prime-strips — i.e., the cessions just discussed — in the various spokes are . identified with one another   ” may be thought of as being equipped -prime-strip “ > D Put another way, the coric with various distinct “holomorphic structures” — i.e., D -prime-strip struc-  -prime-strip structure — corresponding to the various tures that give rise to the D

172 172 SHINICHI MOCHIZUKI spokes. Finally, [cf. the situation discussed in Corollary 3.9, (i)] this diagram sat- isfies the important property of admitting arbitrary permutation symmetries ± ell -Hodge theaters]. Z n - Θ D of the among the spokes [i.e., among the labels ∈ , respectively, with compatible (iv) The constructions of (i), (ii), (iii) are the constructions of Corollary 4.12, (i), (ii), (iii), relative to the functor [i.e., determined by the functorial algorithm] of Proposition 6.7, in the evident sense [cf. the compatibility discussed in Proposition 6.9, (iii)]. additive Finally, we conclude with analogues of Definition 5.5, Corollary 5.6. Definition 6.11. ± (i) We define a Θ -bridge [relative to the given initial Θ-data] to be a poly- ± morphism † Θ ψ ± † † F −→ F  T ± † † † ; T —where F capsule F -group ; is an F = { F F -prime-strip } is a is an T t ∈ T  t l ± -bridge T — that lifts a D -Θ of , indexed by [the underlying set of] F -prime-strips ± Θ † † † φ [cf. Corollary 5.3, (ii)]. In this situation, we shall write → : D D T  ± † F | T | ± † -capsule obtained from the l -capsule l F for the by forming the quotient | T | of T and identifying the {± 1 } of this underlying capsule by the action of T the index set † F indexed by the elements in the fibers of the quotient components of the capsule T ± ± † Θ Θ † via the constituent poly-morphisms of | T T  | [so each con- ψ ψ } = { ∈ T t ± t † stituent F -prime-strip of F positive automorphism is only well-defined up to a | | T [i.e., up to an automorphism such that the induced automorphism of the associated -prime-strip is ], but this indeterminacy will not affect applications of this positive D construction — cf. the discussion of Definition 6.4, (i)]. Also, we shall write †  F T def   T for the = | T |\{ 0 } of nonzero elements of -capsule determined by the subset l ± -bridges | Θ . We define a(n) [iso]morphism of | T ± ± † Θ Θ ‡ ψ ψ ± ± ‡ † † ‡ ′ ( ) F F → F ) −→ −→ F ( T T   to be a pair of poly-isomorphisms ∼ ∼ ‡ † ‡ † ′ ; F F → → F F   T T ± ± ± † Θ ‡ Θ -Θ that lifts a morphism between the associated , D φ φ -bridges .Thereis ± ± ± -bridges. an evident notion of composition of morphisms of Θ ell (ii) We define a Θ -bridge [relative to the given initial Θ-data] ell Θ † ψ ± † †  ± F −→ D T

173 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 173 ± ± † †  ±  † —where is an T ; is a ; D F F = { category equivalent to F } D -torsor t ∈ T t T l F -prime-strips , indexed by [the underlying set of] T —tobea capsule of is a ell † † ± Θ ell  † † -Θ D D D D φ —wherewewrite -bridge → : for the capsule of D - T T ± † [cf. Remark 5.2.1, (i)]. We define a(n) F [iso]morphism prime-strips associated to T ell -bridges of Θ ell ell Θ † Θ ‡ ψ ψ ± ± † ± ±   ‡ ‡ † ′ ( D −→ −→ D ) ) F ( F → T T to be a pair of poly-isomorphisms ∼ ∼ ‡  ‡  ± † ± † ′ D → D F → F ; T T ell ell Θ ell ‡ Θ † φ -bridges , that determines a morphism between the associated φ D -Θ . ± ± ell There is an evident notion of composition of morphisms of Θ -bridges. ± ell (iii) We define a Θ [relative to the given initial Θ-data] to be -Hodge theater a collection of data ell ± † † Θ Θ ψ ψ ± ell ± ± Θ † † ± † †  F ) ←− −→ F =( D HT T  ± ell Θ † ell ± Θ † —where — such that the ψ ; is a Θ is a Θ -bridge -bridge associated data ψ ± ± ± ell † ± ell † Θ Θ { φ } [cf. (i), (ii)] forms a D -Θ , φ -Hodge theater. A(n) [iso]morphism of ± ± ± ell Θ -Hodge theaters is defined to be a pair of morphisms between the respective ± ell -andΘ -bridges that are compatible with one another in the sense associated Θ that they induce the same poly-isomorphism between the respective capsules of ± ell - F -prime-strips. There is an evident notion of composition of morphisms of Θ Hodge theaters. ± ell ± ell Corollary 6.12. (Isomorphisms of Θ Θ Θ -Bridges, - -Bridges, and Relative to a fixed collection of Θ -data : initial Hodge Theaters) set of isomorphisms be- (i) The natural functorially induced map from the ± ± ell ell -Hodge the- (respectively, two Θ -bridges -bridges ; two Θ Θ two tween ± -bridges - Θ ) to the set of isomorphisms between the respective D associated aters ± ell ell -Hodge theaters D -bridges ; associated ) - Θ Θ D associated (respectively, - bijective is . ± ell -bridge and a Θ -full poly-isomor- -bridge, the set of capsule- + (ii) Given a Θ phisms between the respective capsules of F -prime-strips which allow one to glue ± ell ± ell -and Θ -bridges together to form a Θ -Hodge theater forms a the given Θ torsor over the group ) ( ±  V F 1 × } {± l [cf. Proposition 6.6, (iv)]. Moreover, the first factor may be thought of as corre- ± F sponding to the induced isomorphisms of -torsors between the index sets of the l capsules involved. Proof. Assertions (i), (ii) follow immediately from Definition 6.11; Corollary 5.3, (ii) [cf. also Proposition 6.6, (iv), in the case of assertion (ii)]. ©

174 174 SHINICHI MOCHIZUKI Remark 6.12.1. By applying Corollary 6.12, a similar remark to Remark 5.6.1 ell ± ell ± ,andΘ -Hodge theaters -bridges ,Θ -bridges may be made concerning the Θ 6. We leave the routine details to the reader. § studied in the present Relative to a fixed collection of initial Θ -data : Remark 6.12.2. † † † † ± (i) Suppose that ( D -bridge ;write( F D )isaΘ → → F )for   T T ± -bridge [cf. Definition 6.11, (i)]. Then Proposition 6.7 gives the associated D - Θ † †  D D → )fromthis a -bridge ( for constructing a - functorial algorithm Θ D > T ± † † † †  D → ) D ). Suppose that this D -Θ-bridge ( ( D → D -bridge Θ - D  > T T Θ ‡ ‡ ‡ -Θ-bridge associated to a Θ ( arises as the -bridge D  F HT → )[so F J >  ‡ ‡ — cf. Definition 5.5, (ii)]. Then since the portion “ F F ”ofthis → J = T J > [cf. Definition 5.5, (ii), (d)] by the associated completely determined Θ-bridge is -Θ-bridge, one verifies immediately that D ‡ ‡ F F → ” of the Θ-bridge as having been one may regard this portion “ > J functorial algorithm constructed via a similar to the functorial algorithm of Proposition 6.7 [cf. also Definition 5.5, (ii), (d); the discussion of Remark ± † † ( F -bridge → F ). 5.3.1] from the Θ  T Since, moreover, isomorphisms between -bridges are in natural bijective corre- Θ associated -bridges [cf. Corollary Θ - spondence with isomorphisms between the D 5.6, (ii)], it thus follows immediately [cf. Corollary 5.3, (ii)] that isomorphisms are in natural bijective correspondence with isomorphisms be- between Θ -bridges ‡ ‡ F → F ”] considered above. Thus, in -bridges [i.e., “ tween the portions of Θ > J Θ ‡ ‡ ‡ summary, if ( F for which the portion  → HT F )isaΘ -bridge J > ‡ ‡ → ” is obtained via the functorial algorithm discussed above from the F F “ J > ± † † ), then, for simplicity, we shall describe this state of affairs F F → -bridge ( Θ T  by saying that Θ ‡ ‡ ± ‡ )is F →  -bridge HT Θ F glued to the -bridge the Θ ( > J † † functorial algorithm F F ) via the → of Proposition 6.7. ( T  We leave the routine details of giving a more explicit description [say, in the style of the statement of Proposition 6.7] of such functorial algorithms to the reader. A ± D - Θ -bridges and D - Θ similar [but easier!] construction may be given for -bridges . (ii) Now observe that ± ell [cf. Definition 6.11, (iii)] to a ΘNF - -Hodge theater gluing aΘ by Hodge theater [cf. Definition 5.5, (iii)] along the respective associated ± Θ -and via the functorial algorithm of Proposition 6.7 [cf. (i)], -bridges Θ one obtains the notion of a ± ell NF-Hodge theater” “ Θ — cf. Definition 6.13, (i), below. Here, we note that by Proposition 4.8, (ii); Corollary 5.6, (ii), the gluing isomorphism that occurs in such a gluing operation is unique . Then by applying Propositions 4.8, 6.6, and Corollaries 5.6, 6.12, one

175 ̈ INTER-UNIVERSAL TEICHM 175 ULLER THEORY I ell ± analogues NF-Hodge theaters. In a similar may verify of these results for such Θ ell ± ΘNF -Hodge theater to a D - -Hodge theater to obtain a - vein, one may glue a Θ D ± ell NF-Hodge theater” [cf. Definition 6.13, (ii), below]. We leave the routine - Θ “ D details to the reader. Remark 6.12.3. (i) One way to think of the notion of a ΘNF studied in § 4isas -Hodge theater a sort of  -torsors F total space of a local system of l “base space” over a “homotopy” between a number field that represents a sort of bad ∈ V ]. From [i.e., the elliptic curve under consideration at the v and a Tate curve ± ell -Hodge theater studied in the present § 6 this point of view, the notion of a Θ may be thought of as a sort of ±  F total space of a local system of -torsors l   ± F over a similar “base space”. Here, it is interesting to note that these F -and - l l l -torsion points of the elliptic curve under torsors arise, on the one hand, from the consideration, hence may be thought of as discrete approximations of [the geometric portion of] this elliptic curve over a number field [cf. the point of view of scheme-theoretic Hodge-Arakelov theory discussed in [HA- SurI], tempered fundamental § 1.3.4]. On the other hand, if one thinks in terms of the bad   ± -and F V ∈ F , then these -torsors groups of the Tate curves that occur at v l l may be thought of as finite approximations Z ” of the copy of “ Galois group of a well-known tempered covering of the Tate that occurs as the curve [cf. the discussion of [EtTh], Remark 2.16.2]. Note, moreover, that if one ± ell NF-Hodge theaters [cf. Remark 6.12.2, (ii)], then one is, in effect, works with Θ working with both the additive and the multiplicative structures of this copy of Z — although, unlike the situation that occurs when one works with rings , i.e., in which the additive and multiplicative structures are “entangled” with one another in some sort of complicated fashion [cf. the discussion of [AbsTopIII], ell ± NF-Hodge theaters, then each of the additive Remark 5.6.1], if one works with Θ independent fashion [i.e., in the form of and multiplicative structures occurs in an ± ell - and ΘNF-Hodge theaters], i.e., “extracted” from this entanglement . Θ (ii) At this point, it is useful to recall that the idea of a distinct [i.e., from the copy of implicit in the “base space”] “local system-theoretic” copy of Z occurring Z over a “base space” that represents a number field is reminiscent not only of the discussion of [EtTh], Remark 2.16.2, but also of the Teichm ̈ uller-theoretic point of view discussed in [AbsTopIII], § I5. That is to say, relative to the analogy with p -adic Teichm ̈ uller theory , the “base space” that represents a number field corresponds to a hyperbolic curve in positive characteristic, while the “local system-theoretic”

176 176 SHINICHI MOCHIZUKI Z copy of — which, as discussed in (i), also serves as a discrete approximation of the [geometric portion of the] elliptic curve under consideration — corresponds to nilpotent ordinary indigenous bundle over the positive characteristic hyperbolic a curve. (iii) Relative to the analogy discussed in (ii) between the “local system-theoretic” of (i) and the indigenous bundles that occur in p uller theory, Z copy of -adic Teichm ̈ it is interesting to note that the two combinatorial dimensions [cf. [AbsTopIII], Re- ±  -” and mark 5.6.1] corresponding to the additive and multiplicative [i.e., “ F l  ± ell symmetries of Θ -”] -, ΘNF-Hodge theaters may be thought of as corre- F “ l sponding, respectively, to the two real dimensions + ; z a z a , z  →− →  z · + ) · cos( t )+sin( t z − t ) z ) t sin( · cos( , z  → →  · z cos( t t sin( · z z · sin( t ) − ) + cos( t ) ) —where a, t ∈ R ; z denotes the standard coordinate on H — of transformations of the upper half-plane H , i.e., an object that is very closely related to the canonical indigenous bundles that occur in the classical complex uniformization theory of hyperbolic Riemann surfaces [cf. the discussions of Remarks 4.3.3, 5.1.4]. Here, it is also of interest to observe that the above additive symmetry of the upper half-plane is closely related to the coordinate on the upper half-plane determined “classical -parameter” q by the def πiz 2 e = q — a situation that is reminiscent of the close relationship, in the theory of the  ± -symmetry and the Kummer theory F present series of papers, between the l Hodge-Arakelov-theoretic evaluation of the surrounding the on the theta function l at bad primes [cf. Remark 6.12.6, (ii), below; the theory of -torsion points ± ” [cf. Definition 6.1, (v)] with respect fixed V basepoint “ [IUTchII]]. Moreover, the  ± -torsion points in the context of the F to which one considers l -symmetry is rem- l iniscent of the fact that the above additive symmetries of the upper half-plane fix the cusp at infinity . Indeed, taken as a whole, the geometry and coordinate natu- rally associated to this additive symmetry of the upper half-plane may be thought of, at the level of “combinatorial prototypes” , as the geometric apparatus as- sociated to a cusp [i.e., as opposed to a node — cf. the discussion of [NodNon], Introduction]. By contrast, the “toral” multiplicative symmetry of the upper half-plane recalled above is closely related to the coordinate on the upper half-plane that determines a biholomorphic isomorphism with the unit disc def i z − = w + i z — a situation that is reminiscent of the close relationship, in the theory of the  -symmetry and the Kummer theory F present series of papers, between the l [cf. Remark 6.12.6, (iii), below; the theory of number field F surrounding the mod  action of F 5 of the present paper]. Moreover, the § on the “collection of basepoints l Bor un ±  for the l -torsion points” V · V F = [cf. Example 4.3, (i)] in the context of l

177 ̈ INTER-UNIVERSAL TEICHM 177 ULLER THEORY I  F -symmetry is reminiscent of the fact that the multiplicative symmetries of the l act transitively on the the upper half-plane recalled above entire boundary of the . That is to say, taken as a whole, the geometry and coordinate upper half-plane naturally associated to this multiplicative symmetry of the upper half-plane may be thought of, at the level of “combinatorial prototypes” , as the geometric apparatus associated to a node , i.e., of the sort that occurs in the reduction modulo Hecke correspondence p of a [cf. the discussion of [IUTchII], Remark 4.11.4, (iii), (c); [NodNon], Introduction]. Finally, we note that, just as in the case of the ±   -, F -symmetries discussed in the present paper, the only “coric” symmetries, F l l i.e., symmetries common to both the additive and multiplicative symmetries of the {± 1 } ” [i.e., the symmetries upper half-plane recalled above, are the symmetries “ z in the case of the upper half-plane]. The observations of the above z −  z, → discussion are summarized in Fig. 6.4 below. Remark 6.12.4.   ± of Proposition 4.9, (i), the - F -symmetry F (i) Just as in the case of the l l of Proposition 6.8, (i), will eventually be applied, in the theory of the symmetry present series of papers [cf. theory of [IUTchII], [IUTchIII]], to establish an explicit network of comparison isomorphisms log-volumes — associated to the non-labeled relating various objects — such as prime-strips that are permuted by this symmetry [cf. the discussion of Remark  4[cf.the -symmetry studied in § 4.9.1, (i)]. Moreover, just as in the case of the F l discussion of Remark 4.9.2], one important property of this “network of comparison isomorphisms” is that it operates without “label crushing” [cf. Remark 4.9.2, (i)] — i.e., without disturbing the bijective relationship between the set of indices of ∼ → F under the symmetrized collection of prime-strips and the set of labels T ∈ l consideration. Finally, just as in the situation studied in 4, § this crucial synchronization of labels is essentially a consequence of the single connected component — of the global object [i.e., — or, at a more abstract level, the single basepoint  ± † †  ± ell -ΘNF-] D ” in the present ”in § 4] that appears in the [ D -Θ D 6; “ -or D § “ Hodge theater under consideration [cf. Remark 4.9.2, (ii)]. (ii) At a more concrete level, the “synchronization of labels” discussed in (i) is realized by means of the crucial bijections ∼ ∼ ±  † †  † ± † → J → ζ T : LabCusp ) ( D D : LabCusp( ; ) ζ  ± of Propositions 4.7, (iii); 6.5, (iii). Here, we pause to observe that it is precisely the existence of these bijections relating index sets of capsules of D -prime-strips to sets of global [ ± -]label classes of cusps

178 178 SHINICHI MOCHIZUKI ± ell NF-Hodge theaters Classical Θ upper half-plane in inter-universal uller theory Teichm ̈  ± + a Additive F z → z -  , l  →− z symmetry a ( a ∈ R ) symmetry z + def 2 πiz e q = theta fn. evaluated at “Functions” assoc’d add. symm. l -tors. [cf. I, 6.12.6, (ii)] to ± assoc’d cusp V Basepoint single add. symm. to at infinity [cf. I, 6.1, (v)] Combinatorial assoc’d prototype cusp cusp to add. symm. ) sin( − z · cos( t ) t  z Multiplicative - →  , F l ) t sin( · )+cos( z t · cos( t )+sin( t ) z  → ( symmetry symmetry ) R ∈ t z cos( − t sin( · z ) t ) elements of the “Functions” def i z − = number field F assoc’d to w mod + i z mult. symm. [cf. I, 6.12.6, (iii)] ( ) ) ( ) − sin( t ) ) t )sin( t cos( cos( t ± un Bor   F V Basepoints = F V assoc’d · , l l ) t − ) t sin( ) t )cos( t sin( cos( mult. symm. to [cf. I, 4.3, (i)]  boundary of H } { entire nodes of mod p nodes of mod p Combinatorial prototype assoc’d Hecke correspondence Hecke correspondence to [cf. II, 4.11.4, (iii), (c)] [cf. II, 4.11.4, (iii), (c)] mult. symm. z  → z, − z {± 1 } Coric symmetries  ±  -symmetries Fig. 6.4: Comparison of -, F F l l with the geometry of the upper half-plane that distinguishes the finer “combinatorially holomorphic” [cf. Remarks 4.9.1,

179 ̈ INTER-UNIVERSAL TEICHM ULLER THEORY I 179   ± (ii); 4.9.2, (iv)] F F - symmetries of Propositions 4.9, (i); 6.8, (i), from -and l l “combinatorially real analytic” the coarser [cf. Remarks 4.9.1, (ii); 4.9.2, (iv)]  ± -and S - symmetries of Propositions 4.9, (ii), (iii); 6.8, (ii), (iii) — i.e., which S l l admit a compatible bijection between the index sets of the capsules involved do not ± and some sort of [cf. the discussion of Remark 4.9.2, set of [ -]label classes of cusps -]label classes of cusps will play a ± crucial role (i)]. This relationship with a set of [ in the theory of the Hodge-Arakelov-theoretic evaluation of the ́ etale theta function that will be developed in [IUTchII]. (iii) On the other hand, one significant feature of the additive theory of the 6 which does not appear in the multiplicative theory of § present § 4 is the phe- “global ± -synchronization” — i.e., at a more concrete level, the nomenon of † ξ ” that appear in Proposition 6.5, (i), (ii) — between various isomorphisms “ ∈ V . Note that this global ± -indeterminacies that occur at the various v the necessary “pre-condition” [i.e., since the natural addi- -synchronization is a ± ]for | on F F is not compatible with the natural surjection F |  F tive action of l l l l ±  ±  [i.e., corresponding to additive portion the F of F F ]ofthe ⊆ -symmetry l l l ±  -symmetry plays the crucial F Proposition 6.8, (i). This “additive portion” of the l [cf. the discussion zero and nonzero elements of F role of allowing one to relate the l of Remark 6.12.5 below]. † † ζ ξ ’s” ’s” discussed in (ii) and the “ (iv) One important property of both the “ discussed in (iii) is that they are constructed by means of functorial algorithms ± ell -or D -ΘNF-Hodge theater [cf. Propositions D intrinsic structure -Θ from the of a 4.7, (iii); 6.5, (i), (ii), (iii)] — i.e., not by means of comparison with some fixed reference model [cf. the discussion of [AbsTopIII], § I4], such as the objects crucial constructed in Examples 4.3, 4.4, 4.5, 6.2, 6.3. This property will be of when, in the theory of [IUTchIII], we combine the theory developed in importance log-shells developed in [AbsTopIII]. the present series of papers with the theory of Remark 6.12.5.  ±  -symmetry of § 4andthe F - F (i) One fundamental difference between the l l in the § inclusion of the zero element ∈ F symmetry of the present 6 lies in the l means, symmetry under consideration. This inclusion of the zero element ∈ F l network of comparison isomorphisms [cf. Remark in particular, that the resulting 6.12.4, (i)] “zero-labeled” prime-strip to the various “nonzero- allows one to relate the prime-strips , i.e., the prime-strips labeled by nonzero elements labeled”  [or, essentially equivalently, ∈ F ]. F ∈ l l  ± -symmetry allows one to F Moreover, as reviewed in Remark 6.12.4, (ii), the l “com- relate the zero-labeled and non-zero-labeled prime-strips to one another in a binatorially holomorphic” fashion, i.e., in a fashion that is compatible with the † ± ζ ”] with various sets of global -label classes of various natural bijections [i.e., “ cusps . Here, it is useful to recall that evaluation at [torsion points closely related to] the zero-labeled cusps [cf. the discussion of “evaluation points” in Example 4.4, (i)] plays an important role in the theory of normalization of the ́ etale theta function

180 180 SHINICHI MOCHIZUKI “of standard type” — cf. the theory of ́ etale theta functions , as discussed in [EtTh], Theorem 1.10; the theory to be developed in [IUTchII]. ±  § 6 has the ad- -symmetry of the theory of the present (ii) Whereas the F l that it allows one to relate zero-labeled and non-zero-labeled prime-strips, vantage “insulate” it has the [tautological!] disadvantage that it does not allow one to . with the zero-labeled prime-strip confusion the non-zero-labeled prime-strips from This issue will be of substantial importance in the theory of Gaussian Frobenioids [to be developed in [IUTchII]], i.e., Frobenioids that, roughly speaking, arise from the theta values 2 j } q { j v . at the non-zero-labeled evaluation points [cf. the discussion of Example 4.4, (i)] Moreover, ultimately, in [IUTchII], [IUTchIII], we shall relate these Gaussian Frobe- nioids to various global arithmetic line bundles on the number field F . This will and the multiplicative require the use of both the additive structures on the number § 5. field; in particular, it will require the use of the theory developed in (iii) By contrast, since, in the theory of the present series of papers, we shall not be interested in analogues of the Gaussian Frobenioids that involve the zero- labeled evaluation points, we shall not require an “additive analogue” of the portion [cf. Example 5.1] of the theory developed in § 5 concerning global Frobenioids. Remark 6.12.6.  -symmetry of § 4andthe F (i) Another fundamental difference between the l  ± F -symmetry of the present § 6 lies in the geometric nature of the “single base- l ±  -symmetry. That F [cf. the discussion of Remark 6.12.4] that underlies the point” l ∼ ell ± labels ∈ T is to say, the various D -]Θ that appear in a [ → -Hodge theater cor- F l ± ell -Hodge throughout -]Θ D respond — the various portions [e.g., bridges] of the [ single copy [i.e., connected component] of theater — to collections of cusps in a ±  v ∈ V ; these collections of cusps are permuted by the F ” at each -symmetry D “ v l ell -bridge [cf. Proposition 6.8, (i)] without permuting the collection of D -]Θ of the [ ± V ⊆ ( ( K )) [cf. the discussion of Definition 6.1, (v)]. This contrasts V valuations sharply with the arithmetic nature of the “single basepoint” [cf. the discussion  -symmetry of § 4, i.e., in the sense that of Remark 6.12.4] that underlies the F l   F the -translates of -symmetry [cf. Proposition 4.9, (i)] F the various permutes l l ± ± un Bor )) [cf. Example 4.3, (i); Remark 6.1.1]. ⊆ ⊆ V = ( V V ( K V  ± -symmetry of (ii) The of the geometric F nature of the “single basepoint” l ± ell -Hodge theater [cf. (i)] is more suited to the theory of the D -]Θ a[ of the ́ etale theta function Hodge-Arakelov-theoretic evaluation existence of a “single basepoint” cor- to be developed in [IUTchII], in which the bad central ”for v ∈ V plays a D responding to a single connected component of “ v . role  - nature of the “single basepoint” of the F (iii) By contrast, the arithmetic l symmetry of a [ D -]ΘNF-Hodge theater [cf. (i)] is more suited to the

181 ̈ INTER-UNIVERSAL TEICHM 181 ULLER THEORY I of the [cf. Example 5.1] explicit construction number field F mod — i.e., to the construction of an object which is invariant with respect to the ∼  / ( C ) ) → F Aut -symmetries that appear in the discussion of Example C Aut(  K K l 4.3, (iv). That is to say, if one attempts to carry out a similar construction to  ± that appears D the construction of Example 5.1 with respect to the copy of ell -bridge, then one must sacrifice the crucial ridigity with respect to -]Θ D in a [ ∼  ±  ±  D Aut( D [cf. Definition 6.1, (v)] that arises from the struc- / ) Aut → F ) ( ± l ell -bridge [cf. Example 6.3; Definition 6.4, (ii)]. -]Θ ture [i.e., definition] of a [ D  -rigidity, then one no longer has a situation in F Moreover, if one sacrifices this l ” which the symmetry under consideration is defined relative to a single copy of “ D v . In par- ∈ V , i.e., defined with respect to a “single geometric basepoint” v at each  no longer -rigidity, the resulting symmetries are ticular, once one sacrifices this F l with the theory of the Hodge-Arakelov-theoretic evaluation of the ́ etale compatible theta function to be developed in [IUTchII] [cf. (ii)]. (iv) One way to understand the difference discussed in (iii) between the global   ± D D , -]ΘNF-Hodge theater ]ofa[ [i.e., the portions involving copies of portions D ± ell -Hodge theater is as a reflection of the fact that whereas the Borel and a [ D -]Θ subgroup {( )} ∗∗ ( F ) ⊆ SL 2 l 0 ∗ ) [cf. the discussion of Example 4.3], the ( F “semi- normally terminal is SL in l 2 unipotent” subgroup {( )} ∗ ± 1 ( F ) ⊆ SL 2 l 1 ± 0  ±  ± [which corresponds to the subgroup Aut ⊆ ( Aut( D D ) ) — cf. the discussion ± ). F ( SL fails to be normally terminal of Definition 6.1, (v)] in 2 l ± ell NF-Hodge theater [cf. Remark D (v) In summary, taken as a whole, a [ -]Θ 6.12.2, (ii)] may be thought of as a sort of “intricate relay between geometric and arithmetic basepoints” that allows one to carry out, in a consistent fashion, both Hodge-Arakelov-theoretic evaluation of the ́ etale theta (a) the theory of the to be developed in [IUTchII] [cf. (ii)] and function in Example 5.1 [cf. (b) the F explicit construction of the number field mod (iii)]. [cf. Remark 6.12.3], Z finite approximation of as a Moreover, if one thinks of F l  ± then this intricate relay between geometric and arithmetic — or, alternatively, F l  additive! ]- and F multiplicative! ]- basepoints — may be thought of as a [i.e., [i.e., l sort of global combinatorial resolution of the two combinatorial dimen- sions — i.e., additive and multiplicative [cf. [AbsTopIII], Remark 5.6.1] — of the ring Z .

182 182 SHINICHI MOCHIZUKI computational point of view Finally, we observe in passing that — from a [cf. the as a “good approxima- theory of [IUTchIV]!] — it is especially natural to regard F l tion” of is , as is indeed the case in the situations when Z l “sufficiently large” 4 [cf. also Remark 3.1.2, (iv)]. discussed in [GenEll], § ] ] [ [  } {F bad − 1 < 0 <...< l − <... 1 ∈ V v v .   . < <... ± glue glue Θ Θ 0 , "} ⇑ > = ⇐⇑ φ φ ⇒{  ± ( ) )(  − l − <...< 0 1 < 1 <... } {± 1    < 1 <...

183 ̈ INTER-UNIVERSAL TEICHM 183 ULLER THEORY I ± ell D (ii) We define a NF-Hodge theater [relative to the given initial Θ-data] - Θ ell ± D NF -Θ † HT ± ell -Θ D ± ell † -Hodge theater HT Θ - D triple to be a , consisting of the following data: (a) a D -ΘNF † ΘNF HT - D [cf. Definition 6.4, (iii)]; (b) a [cf. Definition 4.6, -Hodge theater ell ± D -Θ † HT and gluing isomorphism (iii)]; (c) the [necessarily unique!] between D -ΘNF † [cf. the discussion of Remark 6.12.2, (i), (ii)]. HT Frobenioid that appears in a Reference Brief description ± ell NF-Hodge theater Θ non , V Data at When v ∈ V v ∈ of F -prime-strip corresponds I, 5.2, (i) corresponding to each to  ±  Π  O / , / v F v I, 5.5, (ii), (iii); tempered Frobenioid bad at v F V I, 3.6, (a); discussion over the portion of ∈ v v preceding I, 5.4 D at > [non-realified]  global Frobenioid I, 5.5, (i), (iii); F mod corresponding to I, 5.1, (iii) F mod [non-realified]  I, 5.5, (i), (iii); F global Frobenioid I, 5.1, (ii), (iii) corresponding to  ( π )  F D 1 [non-realified]  F global Frobenioid I, 5.5, (i), (iii); corresponding to I, 5.1, (iii)  F ( D ) π  1 ± ell Fig. 6.6: The Frobenioids that appear in a Θ NF-Hodge theater

184 184 SHINICHI MOCHIZUKI Bibliography [Andr ́ e] Y. Andr ́ e, On a Geometric Description of Gal( / )anda p -adic Avatar of Q Q p p ̂ Duke Math. J. 119 (2003), pp. 1-39. GT , [Asada] M. Asada, The faithfulness of the monodromy representations associated with J. Pure Appl. Algebra (2001), pp. certain families of algebraic curves, 159 123-147. atze f ̈ [Falt] G. Faltings, Endlichkeitss ̈ aten ̈ uber Zahlk ̈ orpern, In- ur Abelschen Variet ̈ vent. Math. (1983), pp. 349-366. 73 [FRS] B. Fine, G. Rosenberger, and M. Stille, Conjugacy pinched and cyclically Rev. Mat. Univ. Complut. Madrid 10 pinched one-relator groups, (1997), pp. 207-227. letter to G. Faltings (June 1983) in Lochak, L. Schneps, Geo- [Groth] A. Grothendieck, , metric Galois Actions; 1. Around Grothendieck’s Esquisse d’un Programme London Math. Soc. Lect. Note Ser. 242 , Cambridge Univ. Press (1997). [NodNon] Y. Hoshi, S. Mochizuki, On the Combinatorial Anabelian Geometry of Nodally Hiroshima Math. J. Nondegenerate Outer Representations, (2011), pp. 275- 41 342. [CbTpII] Y. Hoshi, S. Mochizuki, Topics Surrounding the Combinatorial Anabelian Ge- , ometry of Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization RIMS Preprint 1762 (November 2012). [JP] K. Joshi and C. Pauly, Hitchin-Mochizuki morphism, Opers and Frobenius- destabilized vector bundles over curves, Adv. Math. 274 (2015), pp. 39-75. 1 [Kim] M. Kim, The motivic fundamental group of P 0 , , ∞} and the theorem of 1 \{ 161 (2005), pp. 629-656. Invent. Math. Siegel, Algebraic number theory , Addison-Wesley Publishing Co. (1970). [Lang] S. Lang, Univalent Functions and Teichm ̈ [Lehto]O.Lehto, , Graduate Texts in uller Spaces 109 Mathematics , Springer-Verlag (1987). [PrfGC] S. Mochizuki, The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields, J. Math. Sci. Univ. Tokyo 3 (1996), pp. 571-627. [ p Ord] S. Mochizuki, A Theory of Ordinary p -adic Curves, Publ. Res. Inst. Math. Sci. 32 (1996), pp. 957-1151. uller Theory p Foundations of p [ Teich] S. Mochizuki, , AMS/IP Studies -adic Teichm ̈ in Advanced Mathematics 11 , American Mathematical Society/International Press (1999). [ GC] S. Mochizuki, The Local Pro- p Anabelian Geometry of Curves, Invent. Math. p 138 (1999), pp. 319-423. [HASurI] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, Arithmetic Fundamental Groups and Noncommutative Algebra , Proceedings of

185 ̈ INTER-UNIVERSAL TEICHM 185 ULLER THEORY I 70 , American Mathematical Society (2002), Symposia in Pure Mathematics pp. 533-569. [HASurII] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves II, Algebraic Geometry 2000, Azumino Adv. Stud. Pure Math. 36 , Math. Soc. , Japan (2002), pp. 81-114. [AbsAnab] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois Theory and Modular Forms , Kluwer Academic Publishers (2004), pp. 77-122. [CanLift] S. Mochizuki, The Absolute Anabelian Geometry of Canonical Curves, Kazuya Kato’s fiftieth birthday , Doc. Math. 2003, Extra Vol. , pp. 609-640. Publ. Res. Inst. Math. Sci. [GeoAnbd] S. Mochizuki, The Geometry of Anabelioids, 40 (2004), pp. 819-881. Publ. Res. Inst. Math. Sci. [SemiAnbd] S. Mochizuki, Semi-graphs of Anabelioids, 42 (2006), pp. 221-322. [QuCnf] S. Mochizuki, Conformal and quasiconformal categorical representation of hy- perbolic Riemann surfaces, Hiroshima Math. J. 36 (2006), pp. 405-441. Math.J.Okayama [GlSol] S. Mochizuki, Global Solvably Closed Anabelian Geometry, 48 (2006), pp. 57-71. Univ. [CombGC] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. 59 (2007), pp. 455-479. [Cusp] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), pp. 451-539. [FrdI] S. Mochizuki, The Geometry of Frobenioids I: The General Theory, Kyushu J. 62 Math. (2008), pp. 293-400. [FrdII] S. Mochizuki, The Geometry of Frobenioids II: Poly-Frobenioids, Kyushu J. Math. 62 (2008), pp. 401-460. ́ [EtTh] S. Mochizuki, The Etale Theta Function and its Frobenioid-theoretic Manifes- Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. tations, [AbsTopI] S. Mochizuki, Topics in Absolute Anabelian Geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19 (2012), pp. 139-242. [AbsTopII] S. Mochizuki, Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms, J. Math. Sci. Univ. Tokyo 20 (2013), pp. 171-269. [AbsTopIII] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruc- J. Math. Sci. Univ. Tokyo 22 (2015), pp. 939-1156. tion Algorithms, [GenEll] S. Mochizuki, Arithmetic Elliptic Curves in General Position, Math.J.Okayama Univ. 52 (2010), pp. 1-28. [CombCusp] S. Mochizuki, On the Combinatorial Cuspidalization of Hyperbolic Curves, Osaka J. Math. 47 (2010), pp. 651-715. [IUTchII] S. Mochizuki, Inter-universal Teichm ̈ uller Theory II: Hodge-Arakelov-theoretic Evaluation , RIMS Preprint 1757 (August 2012).

186 186 SHINICHI MOCHIZUKI Inter-universal Teichm ̈ uller Theory III: Canonical Splittings of [IUTchIII] S. Mochizuki, 1758 the Log-theta-lattice , RIMS Preprint (August 2012). uller Theory IV: Log-volume Computa- [IUTchIV] S. Mochizuki, Inter-universal Teichm ̈ 1759 (August 2012). tions and Set-theoretic Foundations , RIMS Preprint [MNT] S. Mochizuki, H. Nakamura, A. Tamagawa, The Grothendieck conjecture on Sugaku Expositions 14 the fundamental groups of algebraic curves, (2001), pp. 31-53. [Config] S. Mochizuki, A. Tamagawa, The algebraic and anabelian geometry of config- uration spaces, 37 (2008), pp. 75-131. Hokkaido Math. J. Cohomology of number fields , [NSW] J. Neukirch, A. Schmidt, K. Wingberg, Grundlehren der Mathematischen Wissenschaften 323 , Springer-Verlag (2000). [NS] N. Nikolov and D. Segal, Finite index subgroups in profinite groups, C. R. Math. Acad. Sci. Paris 337 (2003), pp. 303-308. [RZ] Ribes and Zaleskii, Profinite Groups Ergebnisse der Mathematik und ihrer , 3 Grenzgebiete , Springer-Verlag (2000). [Stb1] P. F. Stebe, A residual property of certain groups, Proc. Amer. Math. Soc. 26 (1970), pp. 37-42. [Stb2] P. F. Stebe, Conjugacy separability of certain Fuchsian groups, Trans. Amer. Math. Soc. 163 (1972), pp. 173-188. [Stl] J. Stillwell, Classical topology and combinatorial group theory. Second edition , Graduate Texts in Mathematics 72 , Springer-Verlag (1993). [Tama1] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio Math. 109 (1997), pp. 135-194. [Tama2] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. Res. Inst. Math. Sci. 40 (2004), pp. 1291-1336. [Wiles] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), pp. 443-551. Updated versions of preprints are available at the following webpage: http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html

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