Hecke operators on Hilbert-Siegel modular forms

Abstract. We define Hilbert-Siegel modular forms and Hecke “operators ” acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert-Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of mod-ular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals). 1. Introduction. A Siegel modular form F of degree n over the rationals has a Fourier series supported on even integral symmetric n × n matrices. An even integral symmetric matrix can be interpreted as the matrix for a quadratic form on an even integra