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 twodimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.11752
 Bibcode:
 2018arXiv181211752T
 Keywords:

 Mathematics  Number Theory;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory
 EPrint:
 57 pages, 27 figures