Lifts of projective congruence groups, II

Kiming, Ian

doi:10.1090/S0002-9939-2014-12127-7

Lifts of projective congruence groups, II
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Proc. Amer. Math. Soc. 142 (2014), 3761-3770 Request permission

Abstract:

We continue and complete our previous paper “Lifts of projective congruence groups” concerning the question of whether there exist noncongruence subgroups of $\operatorname {SL}_2(\mathbb {Z})$ that are projectively equivalent to one of the groups $\Gamma _0(N)$ or $\Gamma _1(N)$. A complete answer to this question is obtained: In case of $\Gamma _0(N)$ 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 $\Gamma _1(N)$ 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 $\operatorname {SL}_2(\mathbb {Z})$ that can serve as the basis for experimentation with modular forms on noncongruence subgroups.

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