Logarithmic vector-valued modular forms

Abstract

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho :S{L}_2(\mathbb{Z})\to G{L}_p(\mathbb{C})$ 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 Poincaré 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 $S{L}_2(\mathbb{Z})$.

22 pages