Structure of the module of vector-valued modular forms

TL;DR

This paper proves that the space of holomorphic vector-valued modular forms associated with a representation of the modular group is a free module over classical modular forms, providing structural insights and applications.

Contribution

It establishes the freeness of the module of vector-valued modular forms over the algebra of classical modular forms and explores its functorial properties and applications.

Findings

The module of vector-valued modular forms is free of rank equal to the dimension of the representation.

The structure of the module as a functor is analyzed, revealing its algebraic properties.

Calculations of the Hilbert-Poincaré series for specific cases are provided.

Abstract

Let $V$ be a representation of the modular group $\Gamma$ of dimension $p$. We show that the $\mathbb{Z}$-graded space $\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{H}$ considered as a functor from $\Gamma$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincar\'{e} of $\mathcal{H}(V)$ in some cases.