Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables

— Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to 0{p, q) (resp. U(/>, q)) of degree nq (resp. Hodge type nq, nq) to the space of classical holomorphic Siegel modular forms of weight (p + <?)/2 and genus n (resp. holomorphic hermitian modular forms of weight p + q and genus n). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of7] in terms of periods of T) over certain totally geodesic cycles—generalizing Shintani's solution [21] of a conjecture ofShimura. We then choose T) to be the Poincare dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in the holomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher's Theorem we may now conclude that all such coefficients are zero. This paper is the result of a number of years work [9-14], and was announced in [17]. For some time we have been studying the relationship between two types of cohomology classes for arithmetic quotients of the symmetric spaces attached to orthogonal and unitary groups. The first type of cohomology class has a geometric description as the Poincard dual classes to natural cycles on the above arithmetic quotients. These cycles are themselves unions of arithmetic quotients of totally geodesic subsymmetric spaces associated to smaller orthogonal or unitary groups. They generalize the classical Hurwitz correspondences and the cycles in the Hilbert modular surfaces considered by Hirzebruch-Zagier [8]. The second type of cohomology class has an analytic description in terms of automorphic forms on the above arithmetic quotients constructed using the theta correspondence. This correspondence is realized by an integral transformation with kernel a theta function defined on a product of two locally symmetric spaces. The new development that led to this paper is the discovery of a new method to use the Cauchy* Partially supported by NSF Grant # MCS-82-01660. ** Partially supported by NSF Grant =N= DMS-85-01742. 16 122 STEPHEN S. KUDLA AND JOHN J. MILLSON Riemann equations in the general theory of the theta correspondence. Our method is based on a study of the double complex of relative Lie algebra cohomology with values in the oscillator representation associated to a dual reductive pair. We use such algebraic considerations to show that the B-operator applied to one variable of the kernel is an exact differential form in the other variable. Combining this result with Stokes Theorem, we are able to deduce that the transform^ of closed rapidly decreasing differential forms are holomorphic Siegel or hermitian modular forms. Our idea may be summarized by the following formal calculation which is made precise in Lemma 3.3. All the integrals are with respect to z. 8 (JM^ ̂  ̂ ) =± JM^ ̂  ̂ == JM^ ̂  ̂ : J^e^(T, z) A ^{z) = J^(e^(T, z) A T](^)) == o. The key step 3cp = d^ is the double complex argument alluded to above and the last equality is Stokes' Theorem—here it is necessary that T] is rapidly decreasing. We should perhaps point out that the Howe correspondence on the centers of the universal enveloping algebras ofG and G' (see below) gives only that the above integral is harmonic, i.e., annihilated by D = ~88* +~8*~8 and this does not imply that the integral is annihilated by '9 unless it is in L which is not usually the case. The harmonicity of the integral does not imply the vanishing theorem for its indefinite Fourier coefficients. This vanishing is of critical importance to us. Moreover the correspondence at infinity underlying our correspondence does not always agree with Howe's quotient correspondence. For example if G = 0{2p, 2q) then our correspondence assigns a holomorphic representation of Sp^(R) to the trivial representation of 0{2p, 2q). We will discuss this point in detail in a later work. Our lift should be of significance in arithmetic algebraic geometry since it assigns algebraic objects (holomorphic modular forms) to transcendental objects (cuspidal harmonic forms). In particular if it can be established that our lift is injective one could assign compatible systems of/-adic Galois representations to the above cuspidal harmonic forms. In order to state our first theorem we will need to establish some notation. Let M be a quotient of the symmetric space D of G = 0(/», q) (resp. G = V{p, q)) by an arithmetic subgroup F of standard type (see below). To simplify notation in the following discussion we will assume that the underlying number field is Q^ (resp. imaginary quadratic). The general case is treated in the body of the paper. Let V be the standard module for G and ^(V") denote the Schwartz space of the direct sum of n copies of V. In what follows ( , ) will denote a fixed symmetric (resp. hermitian) form on V of signature (p, q) such that G is the isometry group of ( , ). We also assume p ^ q and n ̂ p. Let G' denote Sp^(R) (resp. U(%, n)) and 5' be the non-trivial 2-fold cover of G (resp. the non-trivial 2-fold cover which is trivial on SU(n, n)). Then G X G' operates on ^(V") by the (restriction of the) oscillator representation and consequently the continuous cohomology group H^(G, ^(V")) is a module for 6'. Let fo be the Lie algebra of the maximal compact subgroup K' of 6' that covers U(n) INTERSECTION NUMBERS OF CYCLES ON LOCALLY SYMMETRIC SPACES 123 (resp. U(n) X U(w)) and po be its orthogonal complement in Qo, the Lie algebra of G', computed using the Killing form. Then the projection G' -> G'/K' identifies po with the tangent space to G'/K' at <?K/. We let p~ denote the subspace of po (x) C which is identified with the anti-holomorphic tangent space to G'/K' at <?K' (see § 6 for the definition of the complex structure on G'/K/). We let f be the complexification of fo and q be the parabolic subalgebra of 9' = QQ ® C given by q == f ® p". We will use the following notation. If m is a Lie algebra, / : m -> C is a homomorphism and U is an m-module, then U^ will denote the subspace of U defined by U^ = { u e U : xu = -^{x) u all x e m }. The next definition will play a very important role in this paper. Definition. — A class 9 £H^(G, ^(V")) which is annihilated by p~ will be said to be holomorphic. The set of all such classes will be denoted H^(G, ^(V*))^. We next observe that there is a natural map erH^G^V^^H^C). Indeed we have the restriction map H'^(G, ̂ (V^) -^ ff(r, ^(V")). But the F-invariant linear functional © : ̂ (V^ -> C (see § 3) induces a map H'(r, .S^V")) -> H'(F, C). The map 9 -> 6(9) is then defined to be the composition of the two previous maps. Now let ^ denote the character of SL' given by ^(^') == det^')"* (resp. ^') =det{k')det-{k)-, where det:V{n) X U{n) -> S are the determinants on the first and second factors). We may associate a 6'-homogeneous line bundle ,Sf^ over G'/K/ to the character /^ above. We now consider the bilinear pairing (( , )) : H;(M, C) x H^G, ^(V")) -^ C(S') given by ((^?))(^)-J^A6W)9). Here we have set a == dim^ M and we have represented O^C^') 9) by a closed {a — i)-form on M and T] by a compactly-supported closed z-form on M. We will describe (( , )) and 6 more concretely in § 3. It is well-known (§ 3) that if 9 e H^-^G, ^(V"))^, then we may identify ((T], 9)) { g ' ) with a harmonic (i.e. annihilated by D above) section of the quotient of -S?̂ by a certain arithmetic lattice F' C G'—we will denote this quotient line bundle by oSf^ also. One of our main points is that if 9 is a holomorphic class, then ((^,9)) { g ' ) is identified with a holomorphic section of JSf^. We note that H:,-(G, ^(V-))^ = H^-^G, ^(V))"n H^G, ^(V»))^. Before stating out first theorem we need to introduce a g'-invariant subspace of the continuous cohomology H^(G, ^(V")). We recall [2, 5.1] that the choice of a 124 STEPHEN S. KUDLA AND JOHN J. MILLSON maximal compact subgroup K C G is equivalent to the choice of a positive definite form ( , )o on V which is a minimal majorant of the given form ( , ) on V of signature (j&, q). We define the Gaussian 90 e ̂ (V") by 9o(^...^J =^^--^o. Definition. — The polynomial Fock space S(V) C ^(V^) is defined to be the space of those Schwartz functions on V" of the form p{v-^, ..., »J 90(^1? • • • ? n)9 where p(v^y . .., yj is a polynomial function on V. Remark. — The term (< polynomial Fock space " is chosen because the image ofS^^) under the intertwining map i o f § 6 from the Schrodinger model of the oscillator representation to the Fock model is the subspace ^(C^) of holomorphic polynomials on C^. The subspace S(V") is invariant under the actions of Q and 3' (the complexified Lie algebras ofG and G'). We recall, given a choice ofK, that the van Est Theorem [4], IX, 5.6, gives a canonical isomorphism v from continuous cohomology to relative Lie algebra cohomology v : H:<(G, ^(V-)) -^ H-(g, K; ^(V-)). Definition. — We say that a class 9 e H^(G, ^(V^) takes values in S^) ifv(9) is in the image of the natural map H-(a, K; S(V)) -> H-(a, K; ^(V-)). We may now state our first theorem. In what follows m = p + q. Theorem 1. — (i) The induced pairing ( ( , ) ) : H:(M, C) x H^(G, ^(V"))^ -> r(JS^) takes values in the holomorphic sections, that is, in the space of Siegel (resp. hermitian) modular forms of weight mf2 (resp. m) and appropriate level. (ii) If Y) eH^(M,C) and 9 eH^-^G, ̂ (V^))^, then all Fourier coefficients a^ of ((Y], 9)) (^') are zero except the positive definite or positive semi-definite ones. Suppose further that 9 takes values in S^^. Then there exist certain totally geodesic cycles Cp on M determined by (3 and independent of 7^ and 9 and associated to smaller orthogonal (resp. unitary) groups such that a^ is the period over Gp of the exterior pro