NOTES ON CLASSICAL SHIMURA VARIETIES

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1 NOTES ON CLASSICAL SHIMURA VARIETIES DONU ARAPURA We work over C in these notes. 1. Abelian varieties An abelian variety is a higher dimensional version of an elliptic curve. So first of all it is a complex torus, i.e. a quotient X = C n /L where L is a lattice. And it embeds into some complex projective space as a complex manifold. Note that Chow s theorem ensures that X is necessarily a projective algebraic variety and the group operations + : X X X and : X X are morphisms. We make a couple of remarks: (1) When working over fields other than C, the last few conditions would be taken as the definition of an abelian variety. In other words, an abelian variety is more correctly defined as a group in the category of projective varieties. (2) A one dimensional torus C/L is always projective. But in higher dimensions this no longer guaranteed. We need to understand what the projectivity condition means in explicit terms. To save time, let us use whatever tools we can to arrive at the answer, and then translate into elementary classical terms at the end. From algebraic geometry, we know that an embedding X P N is determined by the very ample divisor class X (hyperplane) or the very ample line bundle L = O X (1). This has the advantage of giving an object on X which doesn t depend on any external data. The basic topological invariant of the line bundle L is its first Chern class c 1 (L) H 2 (X, Z). Since X is a torus, we can identify H 2 (X, Z) = 2 Hom(L, Z). In other words, c 1 (L) can be viewed as an alternating integer valued pairing on the lattice L. Let us call such a form a polarization if it arises as c 1 (L) with L ample. A characterization essentially goes back to Riemann, and polarizations are also called Riemann forms. Theorem 1.1. An element E 2 Hom(L, Z) is a polarization if and only if the extension of E to a real valued form on C n satisfies E(ix, iy) = E(x, y) and E(x, ix) > 0 when x 0. See [BL, M] for details. The conditions are equivalent to saying that H(x, y) = E(ix, y) + ie(x, y) is positive definite Hermitian. H and E are uniquely determined by the other, so they can be used interchangeably, and are both referred to as the polarization. Since E is an alternating form, it is represented by an integer matrix ( ) 0 D D 0 1

2 2 DONU ARAPURA It follows that det E is a perfect square. E is called principal, if this determinant is 1. Let us work out what the conditions mean in terms of a basis. Let L be a lattice spanned by column vectors ω 1,..., ω 2n C n. Let Ω = (ω 1,... ω 2n ). Assume E is a form on L given by ( ) 0 I E = I 0 with respect to this basis. Proposition 1.2. E gives a polarization (necessarily principal) if and only if the Riemann bilinear relations ΩEΩ T = 0 iωe Ω T > 0 hold Proof. Use [BL, 4.2.1] and substitute E 1 = E. Corollary 1.3. With the classical normalization Ω = (Ω 1, I), the above conditions hold if and only if Ω 1 is symmetric with positive definite imaginary part. The n n Siegel upper half plane H n is the set of symmetric n n matrices with positive definite imaginary part. Corollary 1.4. If Ω H n then C n /ΩZ n + Z n is a principally polarized abelian variety. Conversely, every principally polarized abelian variety arises this way for Ω H n. The matrix is not unique, but we will get to that later. 2. Endomorphism algebras Let X = V/L be an abelian variety. An endomorphism is holomorphic homomorphism f : X X. The set of these maps forms a, generally noncommutative, ring End(X). This ring is not too big because any endomorphism acts faithfully on the lattice L = H 1 (X, Z). It is simpler to first study the finite dimensional Q-algebra End 0 (X) = End(X) Q, and then observe that End(X) is an order in it. Recall that when X is an elliptic curve, there are two possibilities. Either End 0 (X) = Q or it is an imaginary quadratic field. There are many more possibilities in general. First note that: Theorem 2.1. End 0 (X) is a product of matrix algebras over finite dimensional division algebras over Q. Equivalently, it is semisimple. This will be deduced from another result. An abelian variety is isogenous to another if there is a surjective homomorphism from one to the other with finite kernel. Isogeny is an equivalence relation. An abelian variety is called simple if does not contain any nontrivial abelian subvarieties. Theorem 2.2 (Poincaré). Every abelian variety is isogenous to a product of simple abelian varieties. Sketch of proof. If X isn t simple, then it contains a proper abelian subvariety Y = W/W L. Form the orthogonal complement W with respect to a polarization. Set Y = W /W L. Then X is isogenous to Y Y, and both of Y and Y can be assumed to be a product of simple varieties by induction.

3 NOTES ON CLASSICAL SHIMURA VARIETIES 3 To prove the theorem 2.1, write X Y n1 1 Y n2 2..., where Y i are simple and nonisogenous. Then End 0 (X) = Mni (End 0 (Y i )) and End 0 (Y i ) are division algebras because Y i are simple (Schur s lemma). Let X = V/L be an abelian variety with polarization H. The adjoint with respect to H: H(ax, y) = H(x, a y) defines an involution on End(V ). Involution means that it is Q-linear, a = a and (ab) = b a. The algebra End 0 (X) sits naturally inside End(V ). It can be identified with the endomorphisms which preserve the rational lattice L Q = L Q. Theorem 2.3. The subring End 0 (X) End(V ) is stable under the involution. Proof. If a End(L Q ) define a End(L Q ) to be the adjoint with respect to E = ImH i.e. E(ax, y) = E(x, a y). This is defined because E is nonsingular. Given a End 0 (X), it preserves L Q, so we can form a End(L Q ). This coincides with the usual adjoint a End(V ) because ImH(ax, y) = ImH(x, a y). Therefore a preserves the rational lattice L Q, and thus defines an element of End 0 (X). The restriction of to End 0 (X) is called the Rosati involution. It is possible to give a more conceptual description as follows. a End 0 (X) induces and endomorphism a on the dual X = V /(dual lattice), the polarization gives an isogeny λ : X X, and a = λ 1 a λ. There is one other important fact. An involution on a Q or R algebra D is positive if T r(aa ) > 0 for every nonzero a D, where the trace T r of an element can be understood as the trace of its image in the regular representation 1 D End(D). Theorem 2.4. The Rosati involution is positive. We turn to the classification of the simple algebras with positive involution. Simple means that there are no nontrivial two sided ideals. Over R, the classification is easy [S, p 151]. Up to isomorphism, the only simple R-algebras with positive involution are the matrix algebras over R, C, H with the involution given by (conjugate) transpose. For the classification over Q, we restrict to the case where D is a division algebra. This corresponds to the case where X is simple. A quaternion algebra over a field F is a four dimensional simple algebra with F as its centre. Over R, there are only two M 2 (R) (the split case) and H. Theorem 2.5 (Albert). Let D be a finite dimensional division algebra D over Q with centre F and a positive involution. Let F 0 F be the invariant subfield with respect to the involution. Then there are four possibilities. Type I D = F = F 0 is a totally real field. Type II F = F 0 is totally real. D is a quaternion algebra over F which is totally indefinite in the sense R F D splits for every embedding of F R. Type III F = F 0 is totally real. D is a quaternion algebra over F which is totally definite in the sense that R F D is never split. Type IV F 0 is totally real, and F/F 0 is a totally imaginary quadratic extension; in particular it s CM. D is an algebra over F. See [M, pp ] for details and more precise statements. 1 It s more natural and typical to use the so called reduced trace, but I don t think it matters for the positivity statement.

4 4 DONU ARAPURA 3. Moduli of principally polarized abelian varieties Recall that given Ω H n we can construct an abelian variety X Ω = C n /L Ω, L Ω = ΩZ n + Z n principally polarized by ( ) 0 I E = I 0 It is not difficult to see that every principally polarized abelian variety is isomorphic to an X Ω, but the Ω is not unique. The cause of the nonuniqueness is easy to understand. Any linear automorphism of C n preserving E and taking L Ω to L Ω induces an isomorphism X Ω = XΩ. So we need to mod out by this. Recall that given a commutative ring R (e.g. Z, R) we define the symplectic group Sp 2n (R) = { M GL 2n (R) M T EM = E } ( ) A B Lemma 3.1. Given Ω H n and M = Sp C D 2n (R) (AΩ + B)(CΩ + D) 1 H n This defines an action of Sp 2n (R) on H n which is transitive. The stabilizer of ii is {( ) } A B AB T = BA T, AA T + BB T = I = U B A n (R) where the isomorphism is given by sending ( ) A B A + ib B A Corollary 3.2. Thus H n = Sp2n (R)/U n (R). We define A n = Sp 2n (Z)\H n = Sp 2n (Z)\Sp 2n (R)/U n (R) As a set it is it is the set of isomorphism classes of principally polarized abelian varieties of dimension n. The action of Sp 2n (Z) is properly discontinuous, so this set can be given the structure of a complex orbifold (aka V-manifold). By passing to a congruence subgroup Γ(l) = ker[sp 2n (Z) Sp 2n (Z/lZ)] When l 3, we get a free action on H n, so the quotient A n,l = Γ(l)\H n is a complex manifold. This parameterizes abelian varieties together with a level l structure. This is a choice of a basis of the l-torsion points which is symplectic with respect to the Weil pairing. In general, for any congruence group Γ(l) Γ Γ(1), we can interpret points of Γ\H n as abelian varieties with generalized level structures. Using a completely different construction Mumford [GIT] proved that Theorem 3.3 (Mumford). A n,l is the set of complex points of scheme over Spec Z. This is a coarse moduli space for all l and fine if l 3.

5 NOTES ON CLASSICAL SHIMURA VARIETIES 5 We refer to [GIT] for the precise meanings of coarse/fine moduli. Fine is the best possible scenario where there is an actually universal family. A n,l are the basic examples of Shimura varieties. In the Shimura variety literature, one often works with all levels simultaneously by taking an inverse limit. This fits nicely with the adelic viewpoint, but I won t do this here. 4. Other examples Fix a Q-algebra D (not necessarily a division algebra) with positive involution, centre F and -fixed subfield F 0. Let V be a D-module with an alternating Q-bilinear form E : V V Q such that E(bu, v) = E(u, b v), b D, u, v, V For the purpose of constructing moduli, we also choose level structure. We refer to this data as a PEL structure. Fix a PEL structure (D, E,...) and consider the set Sh(D, E,...) of isomorphism classes of abelian varieties (X, E) with an inclusion End 0 (X) D of algebras with involution and an isomorphism H 1 (X, Q) = V compatible with E and the D-module structure. The key problem is to make this into some kind moduli space rather just a set. Here we consider the problem of turning this into an orbifold. This was done by Shimura [S] using a case by case construction with respect to the Albert classification (see also [BL, chap 9]). Here we consider the first case only, which is the simplest. Let D = F = F 0 be a totally real field of degree d over Q. V will be D 2m with standard symplectic form, although its role will be suppressed. We have d distinct embeddings σ i : D R. Fix n = md, and let H = (H m ) d. Let Γ = Sp 2m (O D ), where O D is the ring of integers. This acts on H through the homomorphism σ : Γ (Sp 2m (R)) d induced by (σ i ). The quotient Γ\H can be identified with Sh(D, E,... ) for an appropriate choice of E etc. The special case, where m = 1 is known as a Hilbert modular, or Hilbert-Blumenthal, variety. We will merely be content to exhibit the abelian variety corresponding to each m-tuple Ω = (Ω i ) H. We define the subgroup L Ω d i Cm = C n as the image of OD m Om D under the map (α, β) (Ω i σ i (α), σ i (β)) i=1,...,d One sees that this is a lattice, and the quotient C n /L Ω is an abelian variety of the desired type. The other cases are similar, but more complicated notationally. We just look at one classical example of a Shimura variety for a PEL structure of type IV, called a Picard modular surface [LR]. Let D = F be a imaginary quadratic field with discriminant. Note that D. The involution is conjugation, so F 0 = Q. Let V = D 3 and let E be the imaginary part of a hermitian form H with matrix Let G be the special unitary group of the form H viewed as an algebraic group over Q. Over R, H as signature (2, 1), so G(R) = SU(2, 1). This can be see to act on the complex 2-ball B once we identify it with {[z 1, z 2, z 3 ] P 2 z z 2 2 z 3 2 < 0}. The Picard modular surface is the quotient Γ\B, where Γ G is the subgroup stabilizing the lattice O 3 D V.

6 6 DONU ARAPURA 5. Deligne s axioms I want to conclude by saying something about Deligne s more abstract formulation of Shimura varieties [D1, D2]. A Hodge structure consists of lattice H Z and decomposition H = H Z C = H pq p+q=n such that H qp = H pq. A rational or real Hodge structure is defined as above but with H Z replaced by a rational or real vector space. The number n is called the weight, and the set of tuples with H pq 0 is called the type. A polarization is a pairing E : L L Z such that E(x, Cy) is symmetric positive definite, where C H pq = i q p is the so called Weil operator. Hodge structures of weight n arise in nature as the nth the cohomology of compact Kähler manifold. In particular, if X is a complex torus, then H 1 (X) carries a Hodge structure of type {(1, 0), (0, 1)}. This is reversible. Given a Hodge structure of this type, H H Z + H 10 is a complex torus. With the appropriate notion of morphism we have Proposition 5.1. There is an equivalence of categories between the category of complex tori and the category of Hodge structures of type {(1, 0), (0, 1)}. Under this, abelian varieites correspond to the polarizable Hodge structures. From this point of view, the Siegel upper half plane can be interpreted as the space of Hodge structures of the above type. Let U(1) C denote the unit circle. Given a real Hodge structure H, we have a homomorphism h : U(1) GL(H) where h(z) H pq = z q p. This homomorphism determines the Hodge structure. It s easy to see from this the collection of Hodge structures is stable under standard linear algebra operations 2. Suppose that H H n viewed as polarized Hodge structure. Then the are a number of special features: A. The image of h lies in Sp 2n (C) B. E = End(H) = H H carries Hodge structure with E pq = H p q H p +p,q +q. In particular, the type is {( 1, 0), (0, 0), (0, 1)}. C. The Weil operator C = h( 1) is a Cartan involution. This means that the fixed points under g CḡC 1 is compact subgroup of Sp 2n (C). The compactness follows from the fact that the fixed point group preserves the positive definite form E(, C ). Deligne gave the following more general setup 3. A (connected) Shimura datum consists a semisimple adjoint algebraic group G over Q a homomorphism U(1) G(R) such that S1 The characters of U(1) on Lie(G(C)) are z 1, 1, z. (Compare with B. above.) S2 ad h( 1) is a Cartan involution of G(C). (Compare with C.) S3 G(R) is noncompact, and this holds for all Q-factors as well. The axioms will imply that the G(R) orbit of h will be a hermitian symmetric space D of noncompact type. The quotient of D by an arithmetic group Γ G(Q) 2 In fancier terms, the category of Hodge structures is a Tannakian category. 3 Actually I m using a slight modification due to Milne [Mi, p 44]

7 NOTES ON CLASSICAL SHIMURA VARIETIES 7 will be the corresponding Shimura variety. This contains the previous examples and many more. For the PEL examples, G is a symplectic group or unitary group. References [BL] C. Birkenhake, H. Lange, Complex abelian varieties [D1] P. Deligne, Travaux de Shimura, Sem. Bourbaki (1971) [D2] P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and L-functions (1979) [LR] R. Langlands, D. Ramakrishnan, The zeta functions of Picard modular surfaces (1992) [Mi] J. Milne, Introduction to Shimura varieties [GIT] D. Mumford, Geometric invariant theory [M] D. Mumford, Abelian varieties [S] Shimura, On families of polarized abelian varieties and automorphic functions, Ann Math 1963