Cusps, Congruence Groups and Monstrous Dessins

Abstract We study general properties of the dessins d’enfants associated with the Hecke congruence subgroups Γ 0 ( N ) of the modular group PSL 2 ( Z ) . The definition of the Γ 0 ( N ) as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set Γ 0 ( N ) ∖ PSL 2 ( Z ) as the projective lattices N -hyperdistant from a reference one, and hence as the projective line over the ring Z ∕ N Z . The natural action of PSL 2 ( 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.