Cusps, Congruence Groups and Monstrous Dessins

We study general properties of the dessins denfants 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 $Gamma_0(N)backslashmathrm{PSL}_2(mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $mathbb{Z}/Nmathbb{Z}$. The natural action of $mathrm{PSL}_2(mathbb{R})$ on the lattices defines a dessin denfant 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 denfants associated with the $15$ Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group.