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 ∈ X the Hodge structure on Lie GR induced by the adjoint representation is of type {(1, −1), (0, 0), (−1, 1)}.

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 ∈ 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)\(X × G(Af )/K). Generally this will be a disjoint union of locally symmetric spaces Γ\X+, 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 variety. Let’s also define the Shimura variety at infinite level:

Sh(G, X) = G( )\(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 = C\R 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 ψ 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] ≈ Zˆ × ˆ, 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 ∈ 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 ∈ 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 ∈ 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)\(X × G(Af ) would not change. Further, if we have two pairs (E, η) and (E0, η0) and an isogenyf ∈ 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) \(X × G(Af ))/K / G(Q) \G(Af )/K

∼ det

× × Q>0\Af / det K

× × ˆ × of ShK (G, X) onto a ﬁnite set (note that Q>0\Af = Z is proﬁnite, and det K is open).

ˆ ˆ × Proposition 1.1.2. Suppose that K ⊂ GL2(Z) and det K = Z . Then there is an isomorphism + ShK (G, X) → Γ\X , 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 ∈ G(Q)+, k ∈ K. Map (h, g) to s−1h ∈ X+. Let’s see that this is well-deﬁned: if g = s0k0 with s0 ∈ G(Q)+, k0 ∈ K, then (s0)−1 = γs−1, where γ = s(s0)−1 = k0k−1 ∈ G(Q)+ ∩ K = Γ. This shows that (s0)−1h = γs−1h represents the same class in Γ\X+.

Exercise: work out what happens when det K is smaller than Zˆ × (for ˆ 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, ψ) be a symplectic space over Q: V is a ﬁnite-dimensional vector space and ψ : 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 ∈ GL(V ⊗ k) ψ(gu, gv) = ν(g)ψ(u, v), some ν(g) ∈ k

Thus G comes packaged with a character ν : G → Gm, the similitude character.

Suppose J is a complex structure on VR such that ψ(Ju, Jv) = φ(u, v) for all u, v ∈ VR. As we know, a complex structure is the same thing as a Hodge structure of type {(−1, 0), (0, −1)}. Let h: S → GL(VR) be the associated morphism of real algebraic groups. For a + bi ∈ 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 ∈ C× we have ψ(h(z)u, h(z)v) = ψ((a + bJ)u, (a + bJ)v) = ψ(au, av) + ψ(bJu, bJv) + ψ(bJu, av) + ψ(au, bJv) = |z|2 ψ(u, v) where in the last step we used ψ(au, bJv) = ψ(aJu, −bv) = −ψ(bJu, av). Thus h(z) is a simplectic similitude with factor |z|2. 0 I In the example where ψ(u, v) = utΨv for the matrix Ψ = g , an −Ig 0 example of a complex structure J satisfying ψ(Ju, Jv) = ψ(u, v) is J = Ψ itself. Then h(z) = aI2g + bJ ∈ G(R). When does X satisfy the axioms necessary to deﬁne a hermitian symmetric domain? We need to know that Lie G is of type {(1, −1), (0, 0), (1, −1)} (condition (SV1)). We have

Lie G = f ∈ End V ψ(f(u), v) = −ψ(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,±1} 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 ψ. This just means that ψ(u, Jv) is positive deﬁnite.

The corresponding real form of GR is σ −1 G (R) = g ∈ 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) = ψ(u, Jv) + iψ(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) ±ψ(u, Ju) > 0 for all nonzero u ∈ VR. This is the same as saying that the complex structure h is polarized by ±ψ. If one h ∈ 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 ±ψ. 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 {(−1, 0), (0, −1)}:

5 Proposition 2.2.1. A 7→ H1(A, Z) is an equivalence from AV to the category of polarizable integral Hodge structures of type {(−1, 0), (0, −1)}.

Recall that an integral Hodge structure is a free Z-module VZ together with a Hodge structure on the real vector space VZ ⊗ R. Hodge structures of type {(−1, 0), (0, −1)} are the same as complex structures J on L ⊗ R. For an integral Hodge structure VZ of weight n to be polarizable means there exists a nondegenerate alternating morphism of integral Hodge structures

ψ : VZ × VZ → Z(n) (a polarization) for which the induced Hermitian form

H(v, w) = ψC(v, iw) + iψC (v, w) is deﬁnite on V pq of sign ip−q−k. (Further review: Z(n) is the integral Hodge −n structure on Z of type {(−n, n)}; in terms of the h we have h(z) = |z| on R(n) = Z(n) ⊗ R. The condition that ψ be a morphism of Hodge structures is equivalent to the condition that ψ(V pq,V rs) = 0 unless p = s and q = r. In the case of Hodge structures of type {(−1, 0), (0, −1)}, the weight is −1. An alternating map

ψ : VZ × VZ → Z(−1) is a morphism of Hodge structures exactly when ψ(V −1,0,V −1,0) = 0 and similarly for V 0,−1. −1,0 The corresponding abelian variety is then VC /VZ, so that VZ = H1(A, Z) −1,0 0 and Lie A = VC = VC / Fil VC. Let AV0 be the isogeny category: objects are still abelian varieties over C, but morphisms are deﬁned by

HomAV0 (A, B) = HomAV(A, B) ⊗ Q An isogeny is a morphism A → B which divides n: A → A for some nonzero 0 n ∈ Z; thus isogenies in AV are the same as isomorphisms in AV . 0 Lemma 2.2.2. The functor A 7→ H1(A, Q) is an isomorphism from AV to the category of polarizable rational Hodge structures of type {(−1, 0), (0, −1)}.

6 Proof. Certainly the functor is well-defined. Let’s construct the inverse functor. Let VQ be a polarizable rational Hodge structure of type {(−1, 0), (0, −1)}. Suppose that ψ : VQ × VQ → Q(−1) is the polarization. Let VZ ⊂ VQ be a lattice small enough so that ψ is integral on VZ. Then VZ is a polarizable integral Hodge structure and thus defines an abelian variety A. Suppose VZ and V 0 are two different such lattices, corresponding to A and A0, then there Z exists a lattice V 00 contained in their intersection, and then the resulting A00 Z receives isogenies from both A and A0. Thus A and A0 are isomorphic in AV0. Note that in the lemma, polarizations of the Hodge structure correspond to polarizations of A. If A is an abelian variety over C, we have the rational Tate module Vf A = TA ⊗ , where TA = lim A[N]. This is a free -module of rank 2; note Af ←− Af that it is isomorphic to H1(A, Af ) = H1(A, Q) ⊗ Af . A polarization λ on A induces a polarization on H1(A, Z) and thus on Vf A, which we will still call λ. Suppose as before that (V, ψ) is a fixed symplectic space over Q, and let G = GSp(V ). Recall that X is the set of complex structures on V which are polarized by ±ψ. For K ⊂ G(Af ) we get

ShK (G, X) = G(Q)\(X × G(Af )/K).

Proposition 2.2.3. ShK (G, X) classiﬁes isomorphism classes of triples of the form (A, λ, ηK), where

• A is an object of AV0,

• ±λ is a polarization of A,

• ηK is a K-orbit of Af -linear isomorphims

η : V ⊗ Af → Vf (A)

× which carry ψ onto an Af -multiple of λ. Here, an isomorphism between two triples (A, λ, η) → (A0, λ0, η0) is an iso- mophism A → A0 in AV0 (thus an isogeny between the abelian varieties) which sends λ to a Q×-multiple of λ0 and ηK to η0K.

7 Proof. Suppose (A, λ, ηK) is a given triple. By the lemma, A corresponds to a polarizable rational Hodge structure W = H1(A, Q), and λ corresponds to a ±-polarization of W , which we’ll still call λ. Meanwhile, we have the isomorphism η, which identiﬁes V ⊗ Af with Vf (A) = W ⊗ Af . This shows that V and W are rational symplectic modules of the same dimension. There is only one isomorphism class of these: let

a: V → W be an isomorphism such that λ(av, aw) = ψ(v, w) for all v, w ∈ V . Use a to pull back the Hodge structure on W to a Hodge structure h on V . Since a preserves the symplectic structures, h is polarized by ±ψ and therefore represents an element of X. −1 Let g ∈ GL(V ⊗ Af ) be g(v) = (a ⊗ 1) η(v), where a ⊗ 1: V ⊗ Af → W ⊗ Af is the adelic version of a. The hypothesis that η carries ψ onto a multiple of λ implies that g ∈ G(Af ). The triple (A, λ, ηK) then gets sent to the class of (h, g) in ShK (G, X). Let us check that this is well-deﬁned. If we chose a diﬀerent isomorphism a0 : V → W , then a0 = aγ for some γ ∈ G(Q). Then we would have ended −1 −1 up with the pair (γ h, γ g), which is the same as (h, g) in ShK (G, X). In the opposite direction, if (h, g) is a class in ShK (G, X), then h corresponds to a Hodge structure on V for which ±ψ is a polarization. This gives a pair (A, λ), where A is an abelian variety and λ is a ±-polarization. Then ∼ V = H1(A, Q) and V ⊗ Af = Vf (A). Let η be the composition

g ∼ V ⊗ Af / V ⊗ Af / Vf (A).

Since g is a symplectic similitude, it carries ψ onto a multiple of λ.

3 Shimura varieties of Hodge type

If (G, X) is a general Shimura datum, one might wonder whether ShX (G, X) also classiﬁes abelian varieties with some extra structure.

Deﬁnition 3.0.4. (G, X) is of Hodge type if there exists a symplectic space (V, ψ) over Q and a closed embedding G → G(ψ) = GSp(ψ) which carries X onto X(ψ), where X(ψ) is the space of Hodge structures on V of type {(−1, 0), (0, −1)} for which ψ is a ±-polarization.

8 3.1 Example: A unitary Shimura variety

An√ example of this arises when G is a unitary group over Q. Let F = Q( −d) be an imaginary quadratic ﬁeld, let V0 be a ﬁnite-dimensional F - vector space, and let H : V0 × V0 → F be a nondegenerate hermitian form (relative to complex conjugation on F ). Let G = GU(V0) be the group of unitary similitudes of V0. For a Q-algebra k, we have

G(k) = g ∈ GL(V0 ⊗ k) H(gu, gv) = ν(g)H(u, v) Q where ν(g) is a scalar. Note that ν(g) lies in k (set u = v).

We have GR = GU(HC), where HC is the hermitian form on V0 ⊗F C induced from H. Its isomorphism class only depends on the signature (p, q) of HC. Let V be the underlying Q-vector space of V0, and let ψ : V × V → Q be ψ(v, w) = im H(v, w), √ where im means project onto the −d-coordinate. Then ψ is an alternating form on V . Note that dimQ V = 2 dimF V0. We get an embedding of algebraic groups G → G(ψ), because if g ∈ G, then ψ(gv, gw) = im H(gv, gw) = im H(v, w) = ψ(v, w). What’s the image? Since V0 was an F -vector space, we√ get an endor- mophism α: V → V which corresponds to multiplication by d. Then ev- erything in the image of G(ψ) has to commute with α. The endomorphism α corresponds to a map V ⊗ V ∨ → Q, or (via ψ) a map t: V ⊗ V → Q(−1), which works out as t(v, w) = ψ(v, αw). Such morphisms have an action of G(ψ), by tg(v, w) = ν(g)−1t(gv, gw), and for g to commute with α it is necessary and suﬃcient that tg = t. If g commutes with α and lies in G(ψ), then since √ re H(v, w) = im −dH(w, v) = im H(αw, v) = ψ(αw, v), we ﬁnd that g ∈ G. Thus G ⊂ G(ψ) is exactly the subgroup that preserves the tensor t.

Let τ : F → C be an embedding, and let V0,C = V0 ⊗F,τ C, so that V0,C is a hermitian space under H of signature (p, q). Let e1, . . . , en be an orthogonal basis for V0,C with H(ej, ej) = ±1 (depending on whether j ≤ p or not). Let J be the complex structure on VR coming from the endomorphism

9 √ −1 ∈ C = F ⊗Q R. The underlying real vector space of V0,C is VR, which therefore has basis {ej, Jej}. Then ψ(ej, Jej) = ±1 depending on whether j ≤ p. Now let J 0 be the complex structure on V which satisfies J 2 = −I , R VR JJ 0 = J 0J, and ( 0 Jej 1 ≤ j ≤ p J ej = −Jej p + 1 ≤ j ≤ n. Then J 0 defines a Hodge structure which is polarized by ψ. We have to check 0 that for nonzero v ∈ VR we have ψ(v, J v) > 0. It is enough to check this for 0 the ej: we have ψ(ej,J ej) = 1. Being a Hodge structure on V which is polarized by ψ, J 0 determines a morphism h: S → G(ψ). But since J 0 and J commute, h factors through G. Let X be the G(R)-conjugacy class of h. Then X classifies all complex structures on VR which are polarized by ψ and which commute with J. X is a hermitian symmetric domain (exercise!).

Lemma 3.1.1. X = U(p, q)/U(p) × U(q) is the parameter space for positive p-dimensional subspaces of V0,C (with respect to the hermitian form H).

0 Proof. An element of X is a complex structure J on VR polarized by ψ which commutes with J. Note that J 0J −1 is an involution and therefore is diagonalizable with eigenvalues ±1. To such a J 0 we associate W = V J=J0 . R Then for nonzero v ∈ W we have H(v, v) = ψ(v, Jv) = ψ(v, J 0v) > 0, so that W is positive. Similarly W 0 = V J=−J0 is negative, and V = W ⊕ W 0. R R Since the signature of H is (p, q) we must have dim W = p and dim W 0 = q. 0 Conversely if V0,C = W ⊕W is a decomposition into positive and negative 0 subspaces, then let J be the unique complex structure on VR which is J on W and −J on W 0. The set of such subspaces W is permuted transitively by U(p, q), and the stabilizer of any particular W is U(p) × U(q). The above lemma shows that there is an embedding of hermitian symmetric spaces U(p × q)/U(p) × U(q) → Sp(2n)/U(n) which is compatible with the embedding G → GSp(2n). Now we can give an interpretation of the Shimura variety ShK (G, X).

Proposition 3.1.2. ShK (G, X) classiﬁes equivalence classes of quadruples (A, λ, ι, ηK), where

10 • A is an object of AV0, • λ: A → A∨ is a polarization, • ι: F → End A is an action of F on A up to isogeny, • ηK is a K-orbit of F -linear isomorphisms

η : V ⊗ Af → Vf (A).

These are required to satisfy the following properties:

× 1. η carries ψ onto a Af -multiple of λ.

2. There exists an F -linear isomorphism a: V → H1(A, Q) which carries ψ onto an Q×-multiple of λ. Two quadruples are considered equivalent when there is an F -linear isogeny A → A0 carrying one λ onto a Q×-multiple of the other, and one ηK onto the other. Proof. Let (A, λ, ι, ηK) be such a quadruple. The pair (A, λ) determines a rational Hodge structure W = H1(A, Q) which is polarized by an alternating form; as usual we abuse notation by calling this form λ. W gets the structure of an F -vector space via ι, and the Hodge structure is compatible with it. Using the F -isomorphism a: V → W , we get a Hodge structure h on V ; since a is compatible with the symplectic structures, h is polarized by ψ. Since a was F -linear, h commutes with the action of F ⊗R = C. Therefore h belongs to X. −1 Let g ∈ GL(V ⊗Q Af ) be deﬁned by g(v) = (a⊗1) η(v), where a⊗1: V ⊗ Af → W ⊗Af is the adelic version of a. Since a and η are compatible with the symplectic structures and with F , we see that g ∈ G(Af ). The quadruple (A, λ, ι, ηK) gets sent to the class of (h, g) in ShK (G, X) = G(Q)\(X × G(Af ))/K. The opposite direction is very similar to what appears in the proof of Prop. 2.2.3. We’ll ﬁnish this section by reinterpreting the condition (2) appearing in Prop. 3.1.2. Proposition 3.1.3. Let (A, λ, ι, ηK) be a quadruple of objects as in Prop. 3.1.2 satisfying the following conditions:

11 • The polarization λ and the action ι are required to be compatible, in the sense that for α ∈ F , the diagram

λ A / A∨

ι(α) ι(α) / ∨ A λ A commutes.

× • η carries ψ onto an Af -multiple of λ. Then there exists an F -linear isomorphism a: V → √H1(A, Q) carrying ψ × onto a Q -multiple of λ if and only√ if the action√ of ι( −d) on the complex vector space Lie A has eigenvalues −d and − −d with multiplicities p and q, respectively. Proof. The key fact here is a Hasse principle for hermitian inner product spaces: two such spaces are isomorphic if and only if they are isomorphic over every completion. We have the hermitian inner product space V0. If (A, λ, ι, ηK) is a quadruple, then W = H1(A, Q) has the structure of an F -vector space with an alternating Q-bilinear form λ. The condition on the polarization λ shows that λ(αv, w) = λ(v, αw) for α ∈ F . If we deﬁne H0 : W × W → F by √ √ H0(v, w) = λ(v, −dw) + −dλ(v, w), then H0 is F -linear in v and F -semilinear in w (check this), so that W becomes a hermitian inner product space for F/Q. The existence of η shows that the hermitian inner product spaces V0 and W are isomorphic at every

ﬁnite place. Over the real place, V0,R and WR will be isomorphic if and only if WR has signature (p, q). The real vector√ space WR has one complex structure J coming from ∼ the endomorphism −1 ∈ C = F ⊗ R, with respect to which H(v, w) = ψ(v, Jw) + iψ(v, w) has signature (p, q). But then A gives WR another complex structure (the Hodge structure), with respect to which H0(v, w) = ψ(v, J 0w) + iψ(v, w) is positive deﬁnite. We have Lie A = W J0=i. J and J 0 C commute. 0 0 Suppose that the signature of WR is (p , q ). We have already seen from the proof of Lemma 3.1.1 that dim W J=J0 = p0 (dimension as a complex R vector space using J). We’re done as soon as we observe that W J=J0 = R W J=J0=i = (Lie A)J=i. C

12 3.2 PEL Shimura varities

The general PEL setup involves replacing the imaginary quadratic ﬁeld F/Q with a semisimple Q-algebra B which comes equipped with an involution x 7→ x∗. One also needs a V , which will be a B-module equipped with an alternating form ψ satisfying ψ(bv, w) = ψ(v, b∗w) for b ∈ B. From here one deﬁnes an algebraic group G/Q as the group of B-linear automorphisms of V which preserve ψ up to a scalar. Considering complex structures on V which are polarized by ψ, one gets a hermitian symmetric domain X, and a family of Shimura varieties ShK (G, X). These parametrize abelian varieties with endomorphisms by B. For details, see Milne’s notes.