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Article Dans Une Revue Indag.Math. Année : 2020

Cusps, Congruence Groups and Monstrous Dessins

Résumé

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.
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Dates et versions

hal-01982830 , version 1 (07-11-2022)

Licence

Paternité - Pas d'utilisation commerciale

Identifiants

Citer

Valdo Tatitscheff, Yang-Hui He, John Mckay. Cusps, Congruence Groups and Monstrous Dessins. Indag.Math., In press, 31 (6), pp.1015. ⟨10.1016/j.indag.2020.09.005⟩. ⟨hal-01982830⟩
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