Cycles in Hyperbolic Manifolds of Non-compact Type and Fourier Coefficients of Siegel Modular Forms

Throughout the 1980’s, Kudla and the second named author studied integral transforms Λ from closed differential forms on arithmetic quotients of the symmetric spaces of orthogonal and unitary groups to spaces of classical Siegel and Hermitian modular forms ([11, 12, 13, 14]). These transforms came from the theory of dual reductive pairs and the theta correspondence. In [14] they computed the Fourier expansion of Λ(η) in terms of periods of η over certain totally geodesic cycles under the assumption that η was rapidly decreasing. This also gave rise to the realization of intersection numbers of these ‘special’ cycles with cycles with compact support as Fourier coefficients of modular forms. It is clear from [7],[4] and [6] that the situation is far more complicated when the hypothesis of rapid decay is dropped. The purpose of this paper is to initiate a systematic study of this transform for non rapidly decreasing differential forms η by considering the case for the finite volume quotients of hyperbolic space coming from unit groups of isotropic quadratic forms over Q. We expect that many of the techniques and features of this case will carry over to the more general situation.