Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1

Abstract

Thompson has shown that up to conjugation there are only finitely many congruence subgroups of PSL(2, R) of fixed genus. For PSL(2, Z), Cox and Parryfound an explicit bound for the level of a congruence subgroup in terms of its genus. This result was used by the author and Pauli to compute the congruence subgroups of PSL(2, Z) of genus less than or equal to 24. However, the bound of Cox and Parryapplies only to PSL(2, Z). In this paper a result of Zograf is used to find a bound for the level of any congruence subgroup in terms of its genus. Using this result, a list of all congruence subgroups, up to conjugacy, of PSL(2, R) of genus 0 and 1 is found.

This tabulation is used to answer a question of Conway and Norton who asked for a complete list of genus 0 subgroups, , of PSL(2, R) such that

i.	contains (N) for some N.
ii.	contains the translation z → z + k iff k is an integer.

Thompson has also shown that for fixed genus there are only finitely many subgroups of PSL(2, R) which satisfy these conditions. We call these groups “moonshine groups.” The list of genus 1 moonshine groups is also found. All computations were performed using Magma.

Keywords:

• Congruence subgroups
• moonshine
• genus