Lifts of projective congruence groups, II

TL;DR

This paper classifies when noncongruence subgroups of SL_2(Z) are projectively equivalent to certain congruence groups, providing explicit conditions and expanding the set of known examples for modular forms research.

Contribution

It completes the classification of noncongruence subgroups projectively equivalent to Γ_0(N) and Γ_1(N), detailing precise conditions for their existence.

Findings

Noncongruence subgroups of SL_2(Z) exist for Γ_0(N) iff N not in {3,4,8} and N divisible by 4 or an odd prime ≡ 3 mod 4.

Noncongruence subgroups of SL_2(Z) exist for Γ_1(N) iff N > 4.

Provides explicit examples of noncongruence subgroups for experimentation with modular forms.

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 $\Gamma_0(N)$ 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 $\SL_2(\Z)$ that can serve as basis for experimentation with modular forms on noncongruence subgroups.