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 Parry found 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 Parry applies 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, $\overline G$, of $\PSL(2,\R)$ such that

(i) $\overline G$ contains $\overline\Gamma_0(N)$ for some $N$.

(ii) $\overline G$ contains the translation $z\mapsto z+k$ 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.