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Congruence subgroups of groups commensurable with PSL(2,) of genus 0 and 1. (English) Zbl 1099.11022

Thompson has proved that, up to conjugation, there are only finitely subgroups of Λ=PSL 2 () containing a (non-trivial) principal congruence subgroup of the modular group PSL 2 () of a fixed genus. (The genus of a (discrete) subgroup G of Λ is the genus of the Riemann surface 2 /G, where 2 is hyperbolic 2-space.)

This paper provides a detailed version of Thompson’s theorem for the two lowest genus values. They compile a complete list of all congruence subgroups of Λ of genus 0 and 1. The proofs involve computations using Magma.

The results answer a question of Conway and Norton by listing all the so-called “moonshine groups”, i.e. the genus zero subgroups G of Λ which contain (i) a subgroup of the type Γ 0 (N) and (ii) translations only defined by rational integers.

The authors discuss the overlap between their results and those of K. S. Chua and M. L. Lang in “Congruence subgroups associated to the Monster” [Exp. Math. 13, No. 3, 343–360 (2004), see Zbl 1099.11018 above].

11F06Structure of modular groups and generalizations
11F03Modular and automorphic functions
30F35Fuchsian groups and automorphic functions
11F22Relationship of automorphic forms to Lie algebras, etc.
20H05Unimodular groups, congruence subgroups (matrix groups)