Hecke operators on Hilbert-Siegel modular forms

TL;DR

This paper introduces Hilbert-Siegel modular forms and defines Hecke operators acting on them, establishing their action on Fourier coefficients and extending known results from classical Siegel modular forms to a broader setting.

Contribution

It defines Hilbert-Siegel modular forms, identifies natural isomorphisms between their spaces, and determines the action of Hecke operators on their Fourier coefficients, generalizing previous results.

Findings

Hecke operators act on Fourier coefficients via explicit formulas.

Identification of natural isomorphisms between spaces of Hilbert-Siegel modular forms.

Extension of Hafner and Walling's results to the Hilbert-Siegel setting.

Abstract

We define Hilbert-Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms, 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 we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of a 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).