On Hecke theory for Hermitian modular forms

TL;DR

This paper develops the Hecke theory for Hermitian modular forms over imaginary quadratic fields, showing the algebra's commutativity and its structure, and characterizing associated Eisenstein series.

Contribution

It extends Hecke theory to Hermitian modular forms for arbitrary class number, revealing algebraic structure and connections to Eisenstein series.

Findings

Hecke algebra is commutative for Hermitian modular forms

Inert part of the Hecke algebra resembles Siegel case

Characterization of associated Siegel-Eisenstein series

Abstract

In this paper we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a structure analogous to the case of the Siegel modular group and coincides with the tensor product of its $p$-components for inert primes $p$. This leads to a characterization of the associated Siegel-Eisenstein series. The proof also involves Hecke theory for particular congruence subgroups.