Representation growth of Fuchsian groups and modular forms

TL;DR

This paper investigates the asymptotic behavior of the number of group homomorphisms from cocompact Fuchsian groups to general linear groups over finite fields, revealing connections to modular forms and providing explicit growth formulas.

Contribution

It establishes an asymptotic formula for the growth of homomorphisms from Fuchsian groups to GL(n,q), linking the growth to modular forms and Puiseux series expansions.

Findings

Asymptotic formula for | ext{Hom}(\Gamma, ext{GL}_n(q))|

Expression of the coefficient c_{q,n} as a Puiseux series in 1/q

Connection between the series and modular forms of half-integral weight

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.