# Cusps, Congruence Groups and Monstrous Dessins

## TL;DR

This paper explores the properties of dessins d'enfants linked to Hecke congruence subgroups, revealing their combinatorial structure and connections to modular curves and Moonshine phenomena.

## Contribution

It provides a new interpretation of $ ext{PSL}_2( ext{R})$ actions on lattices and tabulates dessins for genus zero subgroups related to Moonshine.

## Key findings

- Characterization of dessins d'enfants for $ ext{PSL}_2( ext{R})$ quotients
- Interpretation of quotient sets as projective lines over $ ext{Z}/N ext{Z}
- Tabulation of dessins for 15 genus zero subgroups

## Abstract

We study general properties of the dessins d'enfants associated with the Hecke congruence subgroups $\Gamma_0(N)$ of the modular group $\mathrm{PSL}_2(\mathbb{R})$. The definition of the $\Gamma_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $\Gamma_0(N)\backslash\mathrm{PSL}_2(\mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $\mathbb{Z}/N\mathbb{Z}$. The natural action of $\mathrm{PSL}_2(\mathbb{R})$ on the lattices defines a dessin d'enfant structure, allowing for a combinatorial approach to features of the classical modular curves, such as the torsion points and the cusps. We tabulate the dessins d'enfants associated with the $15$ Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group.