Triangular modular curves of low genus

Abstract

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy ${\text{PGL}}_2({\mathbb{F}}_q)$ or ${\text{PSL}}_2({\mathbb{F}}_q)$. We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.