Representation growth of Fuchsian groups and modular forms

TL;DR

This work analyzes the asymptotic growth of representations of cocompact oriented Fuchsian groups into GL_n(q). It blends Lang-Weil style counting with detailed character bounds for GL_n-type groups, leveraging Harish-Chandra restriction and Deligne–Lusztig theory to bound irreducible characters, and then derives modular-form–shaped asymptotics for the counts of representations. A key outcome is an explicit dimension formula for Hom(Γ, GL_n(K)) in terms of χ(Γ), the cone of exponents {n/a_i}, and a genus-dependent correction σ_{Γ,n}, valid for large n and q outside a finite exceptional set; the approach further connects representation counting to half-integral weight modular forms via Puiseux expansions in 1/q. The results yield precise asymptotics (and a modular-form interpretation) for the leading growth rate of Hom(Γ, GL_n(q)) and inform the structure of the associated representation varieties, with practical impact for understanding growth patterns in arithmetic and geometric group representations.

Abstract

Let $Γ$ 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 $Γ$. Then $|\mathrm{Hom}(Γ,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-χ(Γ))n^2}$, 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.

Theorems & Definitions (44)

• Conjecture 1
• Theorem A
• Theorem B
• Theorem C
• Proposition 2.1
• Corollary 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• proof