Kronecker's limit formula, holomorphic modular functions and $q$-expansions on certain moonshine groups

TL;DR

This paper investigates the function theory of higher genus moonshine groups, proving that their function fields have generators with integer $q$-expansions and deriving algebraic relations, using modular forms and extensive computational methods.

Contribution

It extends the understanding of moonshine groups beyond genus zero, showing the existence of generators with integer coefficients and minimal poles, and deriving defining polynomial relations.

Findings

Function fields of moonshine groups up to genus three have generators with integer $q$-expansions.

Derived polynomial relations define the underlying algebraic curves.

Determined whether the cusp at infinity is a Weierstrass point for these curves.

Abstract

For any square-free integer $N$ such that the "moonshine group" $\Gamma_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of $\Gamma_0(N)^+$ to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a $q$-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus moonshine groups $\Gamma_0(N)^+$. For all moonshine groups of genus up to and including three, we prove that the corresponding function field admits two generators whose $q$-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether $i\infty$ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.