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Cusps, congruence groups and Monstrous dessins

Tatitscheff, V., He, Y. ORCID: 0000-0002-0787-8380 & McKay, J. (2020). Cusps, congruence groups and Monstrous dessins. Indagationes Mathematicae, doi: 10.1016/j.indag.2020.09.005


We study general properties of the dessins d’enfants associated with the Hecke congruence subgroups Γ0(N) of the modular group PSL2(Z). The definition of the Γ0(N) as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set Γ0(N)∖PSL2(Z) as the projective lattices N-hyperdistant from a reference one, and hence as the projective line over the ring Z∕NZ. The natural action of PSL2(Z) 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 Type: Article
Additional Information: © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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Text - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.

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