Automorphic forms for triangle groups: Integrality properties

TL;DR

This paper classifies primes in denominators of Hauptmodul and modular forms for non-arithmetic triangle groups, revealing a congruence condition linked to the group's generators, with specific results for Hecke groups.

Contribution

It provides a complete classification of primes in denominators for these modular forms, establishing a clear congruence criterion related to group parameters.

Findings

Primes in denominators are characterized by a congruence condition.

For Hecke groups, primes do not appear if and only if they satisfy a specific congruence.

The results connect group structure with arithmetic properties of modular forms.

Abstract

We classify all primes appearing in the denominators of the Hauptmodul $J$ and modular forms for non-arithmetic triangle groups with a cusp. These primes have a congruence condition in terms of the order of the generators of the group. As a corollary we show that for the Hecke group of type $(2,m,\infty)$, the prime $p$ does not appear in the denominator of $J$ if and only if $p\equiv \pm 1\pmod m$.