Mixed Siegel modular forms and special values of certain Dirichlet series

Abstract

Let phi : H-nr --> C be a Siegel modular form of weight l, and let tau (sigma) : H-n --> H-nr be an Eichler embedding, where H-n denotes the Siegel upper half space of degree n. We use the notion of mixed Siegel modular forms to construct the linear map L-phi,L-tau sigma* : Yk+lr(N) --> Y-k(N) of the spaces of Siegel cusp forms for the congruence subgroup Gamma (0)(N) subset of Sp(n, R) and express the Fourier coefficients of the image L-phi,L-tau sigma* f of an element f is an element of Yk+lr(N) under L-phi,L-tau sigma* in terms of special values of a certain Dirichlet series. We also discuss connections between mixed Siegel cusp forms and holomorphic forms on a family of abelian varieties. 2000 Mathematics Subject Classification: 11F30, 11F46, 11F67; 14K10.