Representation growth of Fuchsian groups and modular forms

Abstract

Let $\Gamma$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\Gamma$. Then $|\mathrm{Hom}(\Gamma,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-\chi(\Gamma))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.