A trace formula for vector-valued modular forms

TL;DR

This paper introduces a trace formula for vector-valued modular forms that links the Hilbert-polynomial values to traces of representation matrices, revealing insights into the structure of these forms.

Contribution

It provides a novel trace formula connecting Hilbert-polynomials and representation traces for vector-valued modular forms, enhancing understanding of their algebraic structure.

Findings

Derived a formula expressing Hilbert-polynomials in terms of trace data.

Restricted the weight distribution of free generators in the module.

Established connections between modular form properties and representation theory.

Abstract

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.