Logarithmic vector-valued modular forms

TL;DR

This paper constructs meromorphic logarithmic vector-valued modular forms for integral weights associated with certain representations of the modular group, showing the space of such forms is a free module over classical modular forms.

Contribution

It introduces a method to construct meromorphic logarithmic vector-valued modular forms for all large weights and proves the space of holomorphic forms is a free module of rank p.

Findings

Construction of meromorphic matrix-valued Poincaré series for large weights

Component functions are logarithmic q-series involving powers of log q

The space of holomorphic forms is a free module of rank p over classical modular forms

Abstract

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.