Talk 1.

This first talk introduced the fundamental objects that are studied in Deligne’s paper. As I have mentioned several times, the origin of the work discussed in Deligne’s paper is ultimately the theory of elliptic modular curves, which goes back to the nineteenth century as the theory of elliptic modular functions. In modern language, that old work is reinterpreted to say that elliptic modular curves have canonical models over certain number fields. Here “canonical” can have several meanings, as Milne discusses in the written notes on his talk, and which we shall discuss at length later. Shimura figured out a notion of canonical model that applies to certain other curves (and to certain higher-dimensional varieties) arising in the theory of automorphic functions. He studied first other quotients of the upper half plane by discrete groups that are arithmetically defined, which has the sense that the groups are constructed as automorphisms of some number-theoretic object (e.g. an appropriate quadratic form); later he generalized his work to study (as one might say) quotients of bounded symmetric domains by arithmetically defined discrete groups. Deligne axiomatized and abstracted Shimura’s techniques, following some insightful ideas of Mumford suggesting that Hodge theory could help one understand Shimura’s canonical models.

In Deligne’s axioms, there are two basic pieces of data: first, a reductive algebraic group G over Q; for instance, G might just be GL_{2} over Q, as in the case related to elliptic modular curves. Such an algebraic group automatically comes with a family of arithmetically defined discrete subgroups of G(R). Julia hinted at those groups right at the end of her talk when she wrote Γ = G(Q) ∩ K, where K is an open, compact subgroup of G(A^{f}); those Γ are the discrete groups generalizing GL_{2}(Z) in its relation to elliptic modular curves: if one takes G = GL_{2} and K = GL_{2}(Ẑ), then Γ = GL_{2}(Z).

Second, one needs a generalization of the upper half plane. In Deligne’s approach, this generalization is X, a G(R)-conjugacy class of homomorphisms h : S → G_{/R}, where S is the algebraic group described in §1.2. When Shimura studies these problems, he works much more explicitly, considering instead reductive groups G over Q such that G(R) acts nicely on a Hermitian symmetric space of non-compact type (e.g. the upper half plane). Here, “nicely” means, say, that the stabilizer of a point of the symmetric space in G′(R) is compact (where G′ is the derived group of G as in Deligne’s article). He generally works completely explicitly, and so there is no question of axiomatization of the input data. If you look back at the introductory notes I sent out, you will see that the examples I mentioned (related to quadratic forms) are considered from Shimura’s viewpoint and not from Deligne’s (G, X) viewpoint.

We should make certain we understand the simplest case of such an X in G = GL_{2}. This case is explained in Milne’s notes, but I will repeat it here for convenience. Julia discussed the more-general case of G = GSp_{2g} briefly in her lecture; we shall return to it in the context of abelian varieties. In the GL_{2} case, one obtains the conjugacy class X as follows: in order to keep notation straight, let V be a 2-dimensional vector space over Q and consider GL(V) as an algebraic group over Q. This group is, of course, GL_{2}, if we choose a basis for V. If we choose the structure of a 1-dimensional complex vector space on V_{R}, we have an R-linear action of C on V_{R} and hence of C^{×}. In fact, for any R-algebra A, we get an A-linear action of A ⊗_{R} C on A ⊗_{R} V and hence an A-linear action of (A ⊗_{R} C)^{×} for each A; those actions define a homomorphism S → GL(V)_{R}. I claim that any two homomorphisms derived in this way from complex structures on R ⊗_{Q} V are conjugate by an element of GL(V)(R). This is easy: if J and J′ are two complex structures on R⊗_{Q} V then the one-dimensional complex vector spaces (V_{R})_{J} and (V_{R})_{J′} are isomorphic; this isomorphism of complex vector spaces is an isomorphism of the underlying real vector spaces, which are both just V_{R}, and so the isomorphism is an element of GL(V)(R). If you unwind that long sentence, you will see that conjugation by that element passes between the homomorphisms S → GL(V)_{R} associated to J and J′. As Milne mentioned, one can identify naturally the set of complex structures on V with a space bi-holomorphic to the union of upper and lower half planes: by associating to J its i-eigenspace on V_{C}, one maps the set of complex structures to the space of lines in V_{C}, which is abstractly P^{1}(C); the image consists of those lines that are not stable under complex conjugation, which is abstractly P^{1}(C)\P^{1}(R). (One mild warning: the Hodge structure on V_{R} associated to an h ∈ X is of weight 1; Deligne prefers weight −1 in this case, which requires replacing each h by h^{−1}.)

The datum (G, X) is subject to a set of rather mysterious conditions described in §1.5 (with one condition postponed until §2.1). The motivation for the conditions comes from Hodge theory, and can be found either in Deligne’s Bourbaki talk on the work of Griffiths or in §1 of his second paper on Shimura varieties, contained in the proceedings of the Corvallis conference. For those who know some Hodge theory, the motivations in these papers are pretty convincing, but explaining the background would carry us quite far afield. As Julia mentioned, though, Hodge theory does provide us with a way of understanding X more concretely: if we choose a representation (ρ, V) of G over R, the conjugacy class X yields a family of Hodge structures on V. To be explicit, for each homomorphism h : S → G_{R} in X, one has the composite homomorphism ρ ∘ h : S → GL(V). As Julia explained, such a homomorphism gives a Hodge structure on V: over C, the group S becomes isomorphic to G_{m} ×G_{m} with two canonical characters z and \overline{z}, and V_{C} = ⊕_{p,q}V^{p,q}, where V^{p,q} is the subspace of V_{C} on which S_{C} acts by z^{p}\overline{z}^{q}; furthermore, since ρ∘h is defined over R, one has \overline{V^{p,q}} = V^{q,p}. As h varies in X, the associated Hodge structure on V varies in turn.

The best way to keep track of this variation is the Hodge filtration on V_{C}, which Julia defined. Recall that the p^{th} step of this filtration is F^{p} = ∑_{p′≥p} V^{p′,q}. If V has a pure Hodge structure of weight n (i.e. the only non-zero spaces V^{p,q} have p+q = n—or, equivalently, the weight co-character w : G_{m} → S acts diagonally by x ↦ x^{n}), then V^{p,q} = F^{p} ∩ \overline{F^{q}}. By associating the Hodge filtration on V induced by ρ ∘ h to h ∈ X, one obtains a map from X to an appropriate manifold of flags in V_{C}, reflecting the variation of the Hodge structures on V. (And if we assume V to be pure of weight n, the point in the flag manifold associated to h ∈ X faithfully reflects the Hodge structure as noted.) In the GL_{2} case described above, the flag manifold is simply P^{1}(C). So far, the space X only has a real-analytic structure (as a homogeneous space for the real Lie group G(R)), and the map from X to the flag manifold can be checked to be real analytic. However, there is a unique complex structure on X such that for any choice of (ρ, V) as above, the associated map of X to the flag variety is homolomorphic. This means roughly that the family of Hodge structures on V parameterized by X varies holomorphically. This complex structure can be described explicitly: let h ∈ X and consider the tangent space to X at h. In terms of L = Lie(G_{R}) with its Hodge structure induced by ad ∘h, the tangent space is L/L^{0,0}, since L^{0,0} is the Lie algebra of the centralizer of h in G_{R}. The inclusion L → L_{C} induces L/L^{0,0} → L_{C}/F^{0}(L_{C}), which is an isomorphism (!) of real vector spaces. This isomorphism gives a complex structure on each tangent space of X, which can be checked to have the desired properties.

In the case of (GSp_{2g}, X) that Julia mentioned, one can consider the family of Hodge structures associated to the standard representation of GSp_{2g} on a 2gdimensional space. In the g = 1 case, this amounts essentially to the families of complex structures discussed above. In general, one can relate the Hodge structures that arise in this way to abelian varieties, as we shall see in the subsequent talks.

Using the pair (G, X), one can define a family of associated Shimura varieties, generalizing the elliptic modular curves. The brutal definition is this: let K be an open, compact subgroup of G(A^{f}); then let

KM_{C}(G, X) = G(Q)\(X × (G(A^{f})/K)).

Note, incidentally, that G(A^{f})/K is a discrete set, since K is open in G(A^{f}). One could rewrite this definition as

KM_{C}(G, X) = ∐_{g} Γ_{g}\X,

where g runs over representatives for the double cosets G(Q)\G(A^{f})/K and Γ_{g} = gKg^{−1} ∩G(Q). It is quite important (although we will not discuss it now) first that the quotients Γ_{g}\X, which are a priori only complex-analytic objects, are given (by Baily and Borel) canonical structures as quasi-projective varieties and second that (by the strong approximation theorem) G(Q)\G(A^{f})/K is finite. As a consequence of these two hard facts, the spaces _{K}M_{C}(G, X) are canonically complex algebraic varieties; our ultimate goal is to show that they have very nice models over certain number fields. Keep in mind when comparing this definition with the classical case that X is supposed to be a generalization of the upper half plane (or, perhaps, of the union of upper and lower half planes).

One more comment may be helpful: a typical open, compact subgroup K of G(A^{f}) is of the form ∏ K_{p}, where K_{p} is an open, compact subgroup of G(Q_{p}). In this case, one can write K ∩ G(Q) = ⋂_{p} K_{p} ∩ G(Q), and the subgroup K_{p} can be considered as a congruence condition at p. For example, if G = GL_{2}, one might have as some K_{p} the subgroup of GL_{2}(Z_{p}) of all matrices reducing to the identity modulo p^{n}; then the condition that K_{p} imposes in K ∩GL_{2}(Q) is that the matrices be p-integral and congruent to the identity modulo p^{n}.