What Is...A Stack?, Volume 50, Number 4

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1 WHAT IS... ? A Stack? Dan Edidin 1 coarse moduli space of elliptic curves. As we will is homeomorphic to A Riemann surface of genus 1 1 S S T × = . Therefore, a choice of a point the torus C see below, however, lacks an important addi- to be the origin determines a group structure on tional property and cannot be considered the true the Riemann surface. An elliptic curve is a Riemann moduli space of elliptic curves. surface of genus 1 together with a choice of origin B of elliptic curves over a base space A family π for the group structure. Although all elliptic curves → B X O X → : B such is a fibration with a section − 1 are homeomorphic to the topological group ) b π ( B ∈ b that for every the fiber is an elliptic 1 1 × S S , they may have nonisomorphic complex ( b ) O . Given a family of elliptic curve with origin π structures. A natural question, called the problem X → B → B : j C , define a classifying map curves B − 1 of moduli, is to describe the space of all possible → )) j ( b b π ( C by . Because is the coarse moduli 1 − ′ isomorphism classes of objects of a certain type. ) b ( π )= b ( j j b ( ) if and only if the fibers space, B B − 1 ′ In this article we discuss this question for elliptic b ( ) π and are isomorphic elliptic curves. Moti- curves and explain how we are led to consider the vated by the concept of classifying space in . stacks notion of topology, we require that a moduli space have a moduli space We wish to construct a for elliptic C universal family . This means that if were the curves. Points of the moduli space should corre- moduli space of elliptic curves, there would exist spond to isomorphism classes of elliptic curves. An E→ C a family of elliptic curves such that every elliptic curve can be expressed as a two-sheeted family would be obtained by pulling back the cover of the Riemann sphere branched at the set : j C → B universal family via the map . However, B { , 0 −{ C ∈ λ 1 } ,λ ∞ } , 1 , 0 , . The rational function since every elliptic curve has an involution, there π 3 2 8 − +1) λ ( λ 2 B → X are nontrivial families of elliptic curves such ( λ )= j is an invariant of the curve, which 2 2 ( − λ 1) λ − 1 ( b ) E π B ∈ b E is a fixed for all that , where 0 0 j -invariant. Direct calcu- was classically called the .) The elliptic curve. (Such a family is called isotrivial λ C , 1 }→ C ) 0 → j ( λ −{ , lation shows that the map j : B → C classifying map is the constant map B is a surjective map that is generically a 6:1 cover- ) b → j ( E . This contradicts the existence of a 0 ing. Moreover, two elliptic curves are isomorphic universal family, because the classifying map j -invariant, so if and only if they have the same C → B → B × E B associated to the trivial family 0 the isomorphism class of an elliptic curve is is also constant. determined by a single complex number. A natural To obtain the moduli space of elliptic curves, we C is the moduli space of elliptic conclusion is that must define a new concept, that of a stack. The stack C is called the curves. To an extent this is true: M of elliptic curves , , is a category. Its objects are families of elliptic curves, and a morphism Dan Edidin is professor of mathematics at the University ′ π π e ′ ′ ′ ( → X ) ) → ( X B → B X → X is a pair of maps , of Missouri, Columbia. His email address is [email protected] β ′ B B → satisfying two conditions: . math.missouri.edu N 458 AMS 4 UMBER 50, N OLUME V OTICES OF THE

2 e considered by Deligne and Mumford are mathe- ′ X → X matically related to a class of called champs . gerbes ′ π ↓ π ↓ commutes. 1. The diagram The French word gerbe can be translated either as β ′ B B → “sheaf” or as “stack”. Since the term sheaf was ′ X X 2. is isomorphic to the pullback of via the already in use, perhaps stack was the next logical β ′ choice.) In that paper the authors defined algebraic B B → . map stacks and used them to prove the result of their (Commutative diagrams satisfying condition (2) title. The stacks they defined are now referred to M .) The subcategory of cartesian are called corre- Deligne-Mumford stacks algebraic , and the term as B is called sponding to families over a fixed base stack usually refers to a generalization given by B . Condition (2) says that the fibers over the fiber M. Artin in the early 1970s. During the last ten M of groupoids are , that is, categories where all years stacks have been widely used to prove theo- M morphisms are isomorphisms. More generally, rems in algebraic geometry and related fields. algebraic stack . An algebraic is an example of an For example, in the mid-1990s Kontsevich showed stack is a category fibered in groupoids which has that Gromov-Witten invariants can be defined as a smooth covering by an affine variety, in a way g . Last of genus integrals on the stack of stable maps which we explain below. year Laurent Lafforgue won a Fields Medal for his While this definition may look strange, we proof of the Langlands conjecture for function will see that there is a universal family of elliptic fields. At the heart of the proof is his construction M B . For any variety curves over the category of compactifications of stacks of certain types of B we can construct a similar category : the objects vector bundles, called Drinfeld shtukas , on curves t B T → , and a morphism are maps of varieties defined over finite fields. ′ f t t ′ ′ T → T → ) T → B ) ( B ( T → such that is a map Further Reading ′ f ◦ t t = . It is relatively easy to show that the A nice introduction to algebraic stacks from the B B B determines , so we can identify and category point of view of moduli of vector bundles was X B B → is equivalent . To give a family of curves written by T. Gómez [2]. I wrote an article on the B →M . Let to giving a functor (map of categories) construction of the moduli space of curves [1] C be the category whose objects are families of which also contains an introduction to algebraic elliptic curves with a (nowhere zero) section and stacks. The most comprehensive, and most tech- M . Forgetting the morphisms defined as for nical, treatise on algebraic stacks is the book of C→M . For any family of section defines a functor G. Laumon and L. Moret-Bailly [3]. X → B C elliptic curves , the pullback of via the X →M B M is . Thus, is corresponding map References C→M the moduli space of elliptic curves and [1] D. E , Notes on the construction of the moduli DIDIN is the universal family. Recent Progress in Intersection Theory space of curves, (Bologna, 1997), Birkhäuser Boston, Boston, MA, 2000, Finally, let us see what it means to say that pp. 85–113 (math.AG/9805101). M has a smooth cover by an affine variety. [2] T. L. G Proc. Indian Acad. Sci. ÓMEZ , Algebraic stacks, Consider the Legendre family of elliptic curves Math. Sci. (2001), 1–31 (math.AG/9911198). 111 2 ( x − 1)( y x − λ ) − x λ U = , where varies in [3] G. L Champs Algébriques and L. M AUMON AILLY , -B , ORET 1 C −{ 0 , } U →M is . The corresponding map Springer, Berlin, 2000. a smooth cover in the following sense: given a →M →M U B , the map pulls back to a 12:1 map 1 B of . unbranched covering Current Trends Deligne and Mumford introduced the term stack in their famous paper “The irreducibility of the space of curves of a given genus”, proposing it as an , English substitute for the French word champ which had previously been used in nonabelian cohomology. (The choice of the word “stack” is means “field” in somewhat puzzling, since champ English. One possible explanation is that the “stacks” ... The “WHAT IS ?” column carries short (one- 1 or two-page), nontechnical articles aimed at grad- b ∈ B The number of inverse images of a point uate students. Each article focuses on a single j j ( λ ) -invariant corresponding to an elliptic curve with mathematical object, rather than a whole theory. λ, 1 − λ, 1 /λ,λ/ ( λ − 1) , { is the cardinality of the set − /λ, 1 / (1 1) λ ) } − λ ( times the number of automorphisms Notices The welcomes feedback and suggestions of the curve. This number is always 12. For a general for topics for future columns. Messages may be λ value of the set has six elements and the automorphism [email protected] sent to . group has order 2. PRIL 459 AMS OTICES OF THE N 2003 A

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