Lifts of projective congruence groups, II

Abstract

We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups $Γ_0(N)$ or $Γ_1(N)$. A complete answer to this question is obtained: In case of $Γ_0(N)$ such noncongruence subgroups exist precisely if $N\not\in {3,4,8}$ and we additionally have either that $4\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $Γ_1(N)$ these noncongruence subgroups exist precisely if $N>4$. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of $\SL_2(\Z)$ that can serve as basis for experimentation with modular forms on noncongruence subgroups.