1. Introduction

In order to make Deligne’s paper a bit more comprehensible, I want first to give an informal discussion of just what he is aiming at and second to offer an outline of the contents of each section, which may be helpful for speakers preparing their talks.

The main goal of the paper is to describe the construction of models of certain Shimura varieties over number fields. Most of their features can be understood in the case of Shimura curves, where the algebraic geometry is easier. As complex manifolds, these curves are special finite-volume quotients of the complex upper half plane H by discrete subgroups Γ (or, more precisely, unions of such quotients: they are not generally connected). The subgroups in question are called arithmetic, meaning loosely that they are defined as the Z-points of certain algebraic groups over Q. One such family of groups is the collection of congruence subgroups of SL_{2}(Q), including, for instance, SL_{2}(Z) and its familiar subgroups such as Γ_{0}(N); the associated quotients are then of finite volume but non-compact, and their compactifications (given by adding one point for each cusp) are the elliptic modular curves. All of the other Shimura curves have no cusps, i.e. the quotients Γ\H are already compact and do not require adding any points at infinity. Recall that all compact complex curves are in fact defined by algebraic equations over C, and so we can think of these curves as algebraic varieties to start with.

In concrete terms, the more general Shimura varieties are finite-volume quotients of various generalized upper half planes by (roughly) discrete arithmetic subgroups of their isometry groups. These generalized upper half planes are the bounded symmetric domains of complex analysis (equivalently, non-compact Hermitian symmetric spaces) and include, for instance, the Siegel upper half spaces, which arise in the theory of moduli of abelian varieties over C. In these more general cases, the complex-analytic objects that one starts with are not evidently defined by algebraic equations, and a good deal of work is necessary to prove that they happen to be quasi-projective. Deligne does not discuss these points explicitly in his paper but refers to the work of Baily and Borel, which is the definitive answer to these algebraicity problems.

In order to give a sense of what sorts of groups Γ arise (at least for the Shimura curve case), let me indicate one construction. Let q(x) be an indefinite, integral ternary quadratic form. Then the special orthogonal group of q(x) is (isomorphic to) PSL_{2}(R), and so acts on H. The group Γ of determinant-1, integral orthogonal matrices is a discrete subgroup of PSL_{2}(R). It is a standard (not to say easy) theorem that Γ is large: the quotient of H by Γ has finite volume. Moreover, unless q(x) is isotropic over Q, the quotient is even compact. The arithmetic properties of these quotients turn out to be relatively easy to understand—in the language of Shimura varieties, they are “of PEL type,” meaning that they can be completely understood using the standard algebraic theory of moduli of abelian varieties just as elliptic modular curves can. Here is a related construction that yields many curves that are much more difficult to study: let F be a totally real field and q(x) be an integral ternary quadratic form over F. For each embedding σ : F → R, one obtains a real quadratic form q^{σ}(x). Let’s suppose that for exactly one such σ, the form q^{σ}(x) is indefinite. The special orthogonal group of the real ternary quadratic form q^{σ}(x) is then (isomorphic to) PSL_{2}(R) and so acts on H. Under our assumption about definiteness, the subgroup Γ of F -integral, determinant-1, orthogonal matrices is a discrete subgroup of PSL_{2}(R) such that Γ\H has finite volume; it is even compact except in the case F = Q and q(x) isotropic.

What makes these sorts of groups Γ special is that (in very old-fashioned language) they admit a transformation theory; in geometric terms, there are analogues for these Γ of the elliptic modular curves of higher level, which has as one practical consequence the existence of a theory of Hecke operators. This transformation theory was known to the old masters of automorphic functions (Fricke and Klein among others), and they used special cases for various amusing purposes (e.g. for solving algebraic equations). They did not, however, manage to develop the arithmetic theory of these curves (or, as they would have put it, of the fields of automorphic functions associated to Γ). Recall that in the elliptic modular case (Γ = SL_{2}(Z)) one has refinements beyond just the transformation theory: the elliptic modular curves such as X_{0}(N) have standard models over Q coming from their interpretation as moduli spaces for elliptic curves with level structures; with respect to these standard models, one knows quite a bit about the behavior of points corresponding to CM elliptic curves (e.g. the j-invariant of a CM elliptic curve generates an abelian extension of its field of complex multiplications); finally, thanks to the Eichler-Shimura congruence relations (known in essence to Kronecker), one can compute the zeta functions of the elliptic modular curves in terms of elliptic modular forms of weight 2.

Until Shimura started to study the curves Γ\H, the refined arithmetic theory that exists for elliptic modular curves was not extended to any other cases (excluding some flawed work of Hecke). In the 1960s, however, Shimura set out to study these other cases, using the theory of abelian varieties with complex multiplication that he, Taniyama, and Weil developed in the 1950s. He and his school (including, at the time, Kuga and Satake, then later Miyake and Shih) succeeded in producing striking work vastly generalizing the subtle arithmetic results about elliptic modular curves. Such work has some intrinsic interest (since, for instance, it provides a way of obtaining some abelian extensions of totally imaginary quadratic extensions of totally real fields by adjoining values of transcendental functions), but has also become important in many number-theoretic problems. Two of particular interest: elliptic curves occuring in the Jacobians of Shimura curves generalize the modular elliptic curves, which have recently been quite important; and Shimura varieties can be used to prove algebraicity results for certain special values of automorphic L-functions.

The basic method Shimura uses to study these curves and varieties is to produce families of abelian varieties that they parameterize in the same way that elliptic modular curves parameterize elliptic curves. Even to study the curves Γ\H, one needs quite general abelian varieties, which is why the contents of Brian’s talks on that subject are a necessary prerequisite for reading the paper. One of the fundamental difficulties in Shimura’s technique is that generally these families of abelian varieties can not be described by the algebraic theory of moduli. The crucial property the families have (and the only reason that they can provide useful information about their parameter spaces) is that they contain many CM abelian varieties, which are very well understood thanks to the work of Shimura and others from the 1950s. (In the Shimura-curve case, however, one is lucky: the families of abelian varieties one uses can be handled by the standard algebraic techniques.) A second twist in Shimura’s work (the subject of §6 of Deligne’s article) is that unlike the elliptic modular case, one single family of abelian varieties is often not enough to produce the sort of results that Shimura predicts; instead, one needs an infinite collection of these families in order to deduce the desired properties—this is certainly the case for the Shimura curves mentioned above when F ≠ Q.

Although Shimura’s work was quite important and exciting, it was rather difficult to understand. The dissemination of Shimura’s ideas was aided by Mumford’s suggestion of a more group-theoretic view of the problems Shimura was addressing. Deligne adopted this viewpoint enthusiastically in his Bourbaki talk, which is why we face so many group-theoretic preliminaries before reaching any actual constructions. This formalism has the advantage that the theorems can be proven quickly once the time comes, but it makes for daunting reading. Let’s hope that the highly group-theoretic approach and its advantages become clear during the course of the seminar.

2. An outline of the paper

I’ll give here an outline of the contents of each section of the paper, indicating the main points along the way. Chances are I will have to update this outline as we move along. The paper is somewhat frustrating in that the main results are quickly dispatched in the final section with the aid of everything that comes before; fortunately some of the preliminary material is independently interesting and so it should not be too difficult to motive yourselves to learn. The frame of the argument is due to Shimura and may be found with different notation in any of his papers on the subject. You may find his 1966 ICM talk in particular helpful, since he discusses there the fundamental problems and results informally as they apply to Shimura curves. Chapter 9 of his book Introduction to the Arithmetic Theory of Automorphic Functions also contains a brief discussion of the Shimura-curves case without proofs and some nice examples. Finally, Milne has put an expanded version of his introductory lecture on his web site (www.jmilne.org), which you should read carefully, since it makes some excellent points and is quite clearly written.

It seems to me that the seminar should be basically be yours. I will suggest the main topics of each section and a reasonable order in which to present the components of the paper, but I don’t want you to feel as if you are following a script. For instance, if you can’t fit what I suggest into two hours, then we will just move the essentials of the remaining topics to later talks.

§1. Le langage adélique. This section introduces the Shimura varieties as quasiprojective varieties over C. In reading it, start by focusing on §§1.2-1.3 where the torus S is introduced and related to Hodge structures. In §1.5, Deligne introduces the algebraic groups and related structures that give rise to Shimura varieties (with one additional condition postponed until §2.1). The conditions may be a bit difficult to understand before you know more examples, but those will come when we return to abelian varieties in §4. Keep in mind that the conjugacy class X of homomorphisms h : S → G_{R} is given a canonical holomorphic structure (using the filtration F^{p} of §1.2—take a look at Milne’s explanation of how complex structures are related to P^{1}(C) − P^{1}(R) to understand this); with this structure it plays the same role in the general cases as H (or rather H ∪ −H) does for elliptic modular curves. In §1.8, Deligne defines the Shimura varieties, which are denoted KM_{C}(G, h_{0}) in the paper. It is important to note that these varieties depend both on the group G and on the conjugacy class X of h_{0}. For now don’t worry too much about the contents of §1.7, where Deligne justifies why the Shimura varieties can be considered as schemes and not just as complex-analytic objects. To understand all of this, you should review the case of G = GL_{2} /Q as Milne discussed it in his introductory lecture.

For the moment, you can skim over §§1.9-1.12, but don’t focus on them. The discussion in §§1.13-1.15 is rather technical and should be postponed until it is needed in §5.

§2. Composantes connexes. This section consists of computations describing the connected components of various quotients that came up in §1. It requires many of the group-theoretic preliminaries from §0. Professor Prasad has agreed to give a talk on those group-theoretic issues, and so you should not need to spend too much time deciphering them yourself.

While reading §2, keep in mind some examples from the case of GL_{2} /Q, which Milne discussed in his introductory lecture. (Note that the derived group is SL_{2}, which is simply connected, and so Theorem 2.4 applies.) The behavior of connected components is related, for instance, to the fact that the modular curve X(N) = Γ(N)\H has a canonical model over Q(ζ_{N}) and not over Q.

§3. Modèles. This section contains the basic definitions of the kinds of models of Shimura varieties over number fields that one hopes to construct. The definitions are inspired by the results of §4. The section is in fact mostly definitions—the main theorems are proved in §5—but you should focus on the tricky notion of the reflex field of §3.7 and the definition of the reciprocity law (3.9.1). Both of these notions are best understood starting from the description of CM abelian varieties given in 4.19. To understand the reflex field, you should also work out a few examples beyond, say, G = GL_{2} /Q. You can look at §6 to see the situations that arise there; try to describe their reflex fields explicitly.

§4. Variétés abéliennes. This section is the key to the paper, since the canonical models that Deligne constructs are made with the aid of certain families of abelian varieties. The first part, §§4.1-4.5, consists of some variants on what Brian discussed about polarizations in his talks. In §§4.6-4.15, Deligne explains how to describe abelian varieties over C (with various types of additional structures: polarizations, rings of endomorphisms, and level structures) in terms of linear-algebra data (more precisely, Hodge-theoretic data), in the way that Milne described elliptic curves in his talk. This description can be readily rephrased in terms of the double-coset spaces defined in §1. That translation is crucial and you should understand it well. The section concludes with §§4.16-4.22 in which the algebraic theory of moduli of abelian varieties is used to produce canonical models for the Shimura varieties that arise in §§4.6-4.15. Besides the general moduli theory, Deligne invokes the Shimura-Taniyama reciprocity law that describes CM abelian varieties. This reciprocity law is the engine that drives the entire endeavor, and so you should try to understand what it says.

§5. Techniques de construction. In this section Deligne proves some formal theorems about the existence and uniqueness of canonical models. Here are the main results:

- Theorem 5.1 and Proposition 5.2 show that there are many special points (the points of Shimura varieties that correspond to CM abelian varieties in cases of §§4.6-4.15). The special points are the main tool used to prove things about Shimura varieties in general.

- Proposition 5.3 is used to deduce properties of models of Shimura varieties from properties of their special points. Its immediate consequences are Corollaries 5.4 and 5.5, which guarantee among other things that canonical models are unique if they exist.

-Corollary 5.7 is one of the main tools used to prove the existence of canonical models. It should be strongly emphasized along with Variant 5.9. (The Shimura varieties to which Variant 5.9 applies are called “of Hodge type.”) The proof of Corollary 5.7 is a bit difficult in that it uses the tricky Proposition 1.15, which should be studied at this point. This is the result that allows one to use families of abelian varieties that are not defined by their structures of polarizations, endomorphisms, and levels (which are all that the general moduli theory describes).

-Propositions 5.10 and 5.11 constitute the trick that allows one to construct the canonical models of, for instance, Shimura curves, which are not generally of Hodge type. They apply to the situation mentioned in the introduction in which an infinite collection of families of abelian varieties is necessary to produce a canonical model. (Roughly, this is the situation in which the Shimura varieties are not moduli varieties for anything.)

§6. Modèles étranges. Finally, in this section Deligne constructs some canonical models. He produces models of (in concrete terms) quotients of products of Siegel upper half spaces by arithmetic groups. One particular case covered by the discussion is that of the Shimura curves mentioned above (quotients of H itself). The technique is to use Corollary 5.7 and Propositions 5.10 and 5.11 in a fairly clever way, inspired by Shimura’s approach. Namely, for each totally imaginary quadratic extension of the totally real field F of §6.1 (the F of the introduction, in the case of Shimura curves), one can view the Shimura variety of interest as contained in (a Galois twist of) a Shimura variety of Hodge type. The Shimura variety of Hodge type has a canonical model by Corollary 5.7 (Variant 5.9, in fact), which provides a quasi-canonical model of the original Shimura variety, by another application of Corollary 5.7 combined with Proposition 5.11. Next one uses Proposition 5.10 applied to the quasi-canonical models obtained by varying the totally imaginary quadratic extension of F in order to finish the construction.

3. How to proceed

I should like to start with some of the topics in §1 and then jump to §4, since the results of that section are what inspires everything else in the paper. After we finish §4, we can go back to §2 and start working in order, perhaps with a bit of a detour to clarify some points from §1 using the wealth of examples provided by §4. This schedule means that those interested in speaking on abelian varieties should be prepared to talk pretty soon.

I think it would be a good idea for the speakers to meet with me before their talks in order that I might clarify things, suggest references, and help figure out a reasonable chunk to cover. I’ll try to be in touch with each speaker in turn to schedule such a meeting.

Here is a very tentative division of the topics into talk-sized pieces. I fully expect it to change as we progress, but you can use it as a general guide.

§1: Julia will start by speaking on some of the topics from here, perhaps as far as §1.8 but not beyond.

From here, we will continue with §4: The notation introduced in §§4.1-4.5 can be brought up as needed, and so should not be prepared as a separate talk. The first §4 talk should cover roughly §§4.6-4.15. As noted above, these sections give a dictionary between abelian varieties over C and linear-algebra structures (Hodge theory). The crucial case to explain is that of polarized abelian varieties (without extra endomorphisms), but the speaker should try to indicate what happens more generally (as in 4.11) as well. The discussion should focus on Hodge theory and not on moduli. The second §4 talk should discuss some of what is in §§4.16-4.22. I can talk to the speaker about the best way to handle the material, but at a minimum there should be a discussion of moduli of abelian varieties from an analytic point of view, viz of the Siegel upper half spaces. Some indication of the other generalizations of H that come up in §4 would also be appropriate. The sticky point is deciding how much time to spend on the question of the algebraic theory of moduli, which can be an infinite sink for effort. For the moment, we will plan on just stating some results without indicating how they are proven.

I will give a talk on CM abelian varieties (in particular on 4.19) sometime before we start §3.

§2: At this point, I hope to have Prasad speak on group-theoretic background from §0 to prepare us for the technical results of §2. The student talks on the contents of this section should take one or two weeks of talks to finish.

§3: In addition to giving the basic definitions, the speakers for this section will need to pull together whatever form of the results of §4 has been presented to explain how it gives canonical models. As for the definitions, the speakers should work out some interesting examples of the field E(G, h) so that its (somewhat bizarre) definition is more-or-less intelligible. All this should take at least two weeks.

§5: It will take at least two sessions to make it through §5 and perhaps three. To divide it into three blocks, we should start with Theorem 5.1 and Proposition 5.2; continue with Proposition 5.3 and Corollaries 5.4-5.7; and finish with Propositions 5.10 and 5.11. Note that the proof of Corollary 5.7 requires Proposition 1.15, which we shall have left untouched until this point. It may be worth taking it for granted, depending on how tired we are.

§6: If there has been enough time and willpower to make a thorough discussion of §5, putting everything together should take only one week, especially if we work out only the simpler cases of Theorem 6.4. It may, however, be worth taking two weeks in order to study the details carefully and leisurely enough so that it all sinks in (this is the main result after all). By this point, we may have enough experience to be a bit more careful about some of the technical points that will have been skipped in earlier talks (for instance, §1.7, which I do not expect us to discuss). If there is interest, we can return to those while explaining the proofs of the main theorems.