Representation growth of Fuchsian groups and modular forms

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 |Hom(Γ,GLₙ(q))|~ cq,n q(1-χ(Γ))n², where cq,n is periodic in n. As a function of q, cq,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.