Cusps, Congruence Groups and Monstrous Dessins

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.

57 pages, 27 figures