Triangular modular curves of low genus

TL;DR

This paper introduces a computational method to enumerate low-genus triangular modular curves, which generalize classical modular curves and are linked to Belyi maps with specific monodromy groups.

Contribution

The paper develops a new computational approach to classify low-genus triangular modular curves at prime levels, expanding understanding of their structure and properties.

Findings

Enumeration of Borel-type triangular modular curves for small genus

Identification of curves with specific monodromy groups

Extension of classification methods for modular curves

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.