Representation growth of Fuchsian groups and modular forms

Michael J. Larsen, Jay Taylor and Pham Huu Tiep

Vol. 336 (2025), No. 1-2, 217–247

DOI: 10.2140/pjm.2025.336.217

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 nontrivial torsion element of $\Gamma$. Then $|\mathrm{Hom}<!--FUNCTION\; APPLICATION-->(\Gamma ,{\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(q))|\sim c_{q,n}{q}^{[t](1-\chi (\Gamma ))n^2}$, where $c_{q,n}$ is periodic in $n$. Within a fixed congruence class for $q$ and for $n$, $c_{q,n}$ can be expressed as a Puiseux series in $1∕q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.

To the memory of Gary Seitz

Keywords

Fuchsian groups, modular forms, representation growth, finite fields

Mathematical Subject Classification

Primary: 20H10

Secondary: 11F20, 11F27, 20C15, 20C33, 20G40

Milestones

Received: 14 July 2024

Revised: 4 December 2024

Accepted: 6 January 2025

Published: 26 May 2025