Matrix-valued Hilbert modular forms

TL;DR

This paper introduces a general framework for matrix-valued Hilbert modular forms, extending existing concepts, and demonstrates their unique Fourier expansions with specific examples.

Contribution

It generalizes logarithmic vector-valued modular forms to matrix-valued Hilbert modular forms and establishes their Fourier expansion properties.

Findings

Unique polynomial Fourier expansions for matrix-valued Hilbert modular forms

Construction of explicit examples in particular cases

Extension of modular form theory to matrix-valued settings

Abstract

In this paper we generalize the notion of logarithmic vector-valued modular form in order to give a general definition of matrix-valued Hilbert modular forms. We prove that they admit unique polynomial Fourier expansions and we build examples in some particular cases.