Arithmetic Properties of Fourier-Jacobi Expansions of Automorphic Forms in Several Variables

Abstract

In this paper we establish some basic arithmetic results concerning automorphic forms in several variables. Our results are, in part, analogues of the classical facts for elliptic modular forms, and results of Shimura on Hilbert and Siegel modular forms, [9] and [10]. In essence, this involves relating rationality properties of Fourier-Jacobi coefficients to rationality properties in algebraic geometry. Our automorphic forms are C-valued holomorphic functions on certain Cartesian powers Hnd of Siegel upper half-spaces Hn. We consider appropriate arithmetic groups r acting on Hnd such that r\H,d is noncompact. In particular, we reduce our study to that of congruence subgroups r of reductive linear algebraic groups G such that the derived group G' is almost simple over Q. Further, let K be a maximal compact subgroup of G'(R)+, the connected component of the identity in the real