Structure of the module of vector-valued modular forms

Abstract

Let $V$ be a representation of the modular group $Γ$ 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 $Γ$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincaré of $\mathcal{H}(V)$ in some cases.