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The Formalism of Shimura Varieties

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The Formalism of Shimura Varieties The formalism of Shimura varieties November 25, 2013 1 Shimura varieties: general definition Definition 1.0.1. A Shimura datum is a pair (G; X), where G=Q is a reduc- tive group, and X is a G(R)-conjugacy class of homomorphisms h: S ! GR satisfying the conditions: 1. (SV1): For h 2 X the Hodge structure on Lie GR induced by the adjoint representation is of type f(1; −1); (0; 0); (−1; 1)g. 2. (SV2): The involution ad h(i) is Cartan on Gad. 3. (SV3): G has no Q-factor on which the projection of h is trivial. (If the conditions above pertain to one h 2 X, they pertain to all of them.) For a compact open subgroup K ⊂ G(Af ), define the Shimura variety ShK (G; X) by ShK (G; X) = G(Q)n(X × G(Af )=K): Generally this will be a disjoint union of locally symmetric spaces ΓnX+, where X+ is a connected component of X and Γ ⊂ G(Q)+ is an arithmetic subgroup. By the Baily-Borel theorem, ShK (G; X) is a quasi-projective va- riety. Let's also define the Shimura variety at infinite level: Sh(G; X) = G( )n(X × G( )) = lim Sh (G; X): Q Af − K K This is also a scheme, but not generally of finite type. Note that Sh(G; X) has a right action of G(Af ). 1 1.1 An orienting example: the tower of modular curves It's good to keep in mind the case of modular curves. Let G = GL2 =Q, a b and let X be the conjugacy class of h: a + bi 7! . Then we have a −b a ∼ G(R)-equivariant bijection X = CnR sending h to i. We think of X as the set of complex structures on the rational vector space V = Q2. It so happens that these complex structures are all polarized (up to sign) by a common alternating form on V , but since such a form is unique up to scaling, we can ignore the polarizations in this discussion. The right way to think about Sh(G; X) is in terms of isogeny classes of elliptic curves eqipped with full (adelic) level structure. For an elliptic curve E= , the (adelic) Tate module is TE = lim E[N] ≈ Z^ × ^, and the rational C −N Z (adelic) Tate module is V (E) = lim E[N] ⊗ = H (E; ) ⊗ f − Af 1 Q Q Af N Proposition 1.1.1. Sh(G; X) classifies isogeny classes of pairs (E; η), where E=C is an elliptic curve and η : Af × Af ! Vf (E) 0 0 is an Af -linear isomorphism. (Two pairs (E; η) and (E ; η ) are isogenous if there is an f 2 Hom(E; E0) ⊗ Q carrying η onto η0.) Proof. Let (E; η) be given. Let W = H1(E; Q). Choose any isomorphism afromV ! W . E gives W a complex structure, which gets transferred via a to a complex structure on V , and thus we a point h 2 X. The rational Tate module Vf E is canonically isomorphic to W ⊗Q Af , and the level structure η becomes an isomorphism η : Af × Af ! LQ ⊗Q Af : On the other hand we have the isomorphism a ⊗ 1: Af × Af ! LQ ⊗Q Af : Let g = (a ⊗ 1)−1η. To (E; η) we assign the class of (h; g) in Sh(G; X). Let's see that this is well-defined. If a different isomorphism a0 : V ! W was chosen, then 2 a0 = γa for some γ 2 G(Q). The complex structure h and the element g would both get translated by the same γ, so that the resulting class in Sh(G; X) = G(Q)n(X × G(Af ) would not change. Further, if we have two pairs (E; η) and (E0; η0) and an isogenyf 2 Hom(E; E0) ⊗ Q between them, 0 0 then f induces isomorphisms H1(E; Q) ! H1(E ; Q) and Vf E ! Vf E . If 0 0 we choose a : V ! H1(E ; Q) to be the composite of a with H1(E; Q) ! 0 H1(E ; Q), then the resulting pair (h; g) does not change at all. The opposite direction is an exercise (or see below where it is written up in the context of Siegel modular varieties). Note that ShK (G; X) is disconnected. One way to see this is that we have a projection + + + ShK (G; X) = G(Q) n(X × G(Af ))=K / G(Q) nG(Af )=K ∼ det × × Q>0nAf = det K × × ^ × of ShK (G; X) onto a finite set (note that Q>0nAf = Z is profinite, and det K is open). ^ ^ × Proposition 1.1.2. Suppose that K ⊂ GL2(Z) and det K = Z . Then there is an isomorphism + ShK (G; X) ! ΓnX ; where X+ is the upper half-plane. Proof. We only give the map in one direction. Let (h; g) represent a class in + ShK (G; X). By the assumption on K we have G(Af ) = G(Q) K, so that we can write g = sk with s 2 G(Q)+, k 2 K. Map (h; g) to s−1h 2 X+. Let's see that this is well-defined: if g = s0k0 with s0 2 G(Q)+, k0 2 K, then (s0)−1 = γs−1, where γ = s(s0)−1 = k0k−1 2 G(Q)+ \ K = Γ. This shows that (s0)−1h = γs−1h represents the same class in ΓnX+. Exercise: work out what happens when det K is smaller than Z^ × (for ^ instance, when K is the subgroup of GL2(Z) consisting of matrices congruent to 1 modulo N). 3 2 The Siegel modular variety This is the most direct generalization of the tower of modular curves. But to generalize correctly, we really do need to take care of polarizations, which forces us to work with GSp2n. Let (V; ) be a symplectic space over Q: V is a finite-dimensional vector space and : V × V ! Q is a nondegenerate symplectic form. Let G = GSp(V ) be the group of symplectic similitudes. For a Q-algebra k: × G(k) = g 2 GL(V ⊗ k) (gu; gv) = ν(g) (u; v); some ν(g) 2 k Thus G comes packaged with a character ν : G ! Gm, the similitude char- acter. Suppose J is a complex structure on VR such that (Ju; Jv) = φ(u; v) for all u; v 2 VR. As we know, a complex structure is the same thing as a Hodge structure of type f(−1; 0); (0; −1)g. Let h: S ! GL(VR) be the associated morphism of real algebraic groups. For a + bi 2 C× = S(R), we have h(a + bi)(v) = av + bJv: Lemma 2.0.3. h factors through GR = GSp(V ). Proof. For z = a + bi 2 C× we have (h(z)u; h(z)v) = ((a + bJ)u; (a + bJ)v) = (au; av) + (bJu; bJv) + (bJu; av) + (au; bJv) = jzj2 (u; v) where in the last step we used (au; bJv) = (aJu; −bv) = − (bJu; av). Thus h(z) is a simplectic similitude with factor jzj2. 0 I In the example where (u; v) = utΨv for the matrix Ψ = g , an −Ig 0 example of a complex structure J satisfying (Ju; Jv) = (u; v) is J = Ψ itself. Then h(z) = aI2g + bJ 2 G(R). When does X satisfy the axioms necessary to define a hermitian symmet- ric domain? We need to know that Lie G is of type f(1; −1); (0; 0); (1; −1)g (condition (SV1)). We have Lie G = f 2 End V (f(u); v) = − (u; f(v)) : R R 4 Under the Hodge structure induced by h: S ! GR, this gets an action of ad h(z), namely f 7! h(z)−1 ◦ f ◦ h(z). We have M ± ± Hom(VC;VC) = Hom(V ;V )) {±1;±1g where V + = V −1;0 and V − = V 0;−1. On V ±, h(z) acts as z (resp., z), so h(z) acts on these factors as 1, z=z, and z=z as required. Condition (SV 2) is that ad h(i) = ad J be Cartan on the adjoint group of GR. But this is equivalent to the condition that every representation (or one faithful representation) of GR be h(i)-polarizable. Let's examine the condition that the tautological representation VR is J-polarized by . This just means that (u; Jv) is positive definite. The corresponding real form of GR is σ −1 G (R) = g 2 GSp(V ) g = JgJ C For g to live in here we need gtΨg = ν(g)Ψ. Let H : VC × VC ! C be the hermitian form H(u; v) = (u; Jv) + i (u; v) σ ∼ Then G = U(H)\Sp(VC). For this to be compact, we need H to be definite, which is to say that (for some choice of sign) ± (u; Ju) > 0 for all nonzero u 2 VR. This is the same as saying that the complex structure h is polarized by ± . If one h 2 X has this property, then they all do. For the remainder of the section, let X be the set of complex structures + − + on VR which are polarized by ± . We have X = X [ X , where X is the Siegel upper half-plane. 2.1 The moduli interpretation of the Siegel modular variety 2.2 Quick review of abelian varieties Let AV be the category of abelian varieties over C. By now we know that this category is equivalent to the category of polarizable integral Hodge structures of type f(−1; 0); (0; −1)g: 5 Proposition 2.2.1.
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