Lifts of projective congruence groups, II

Details Lifts of projective congruence groups, II Lifts of projective congruence groups, II

2011-12-29

arxiv.org/abs/1112.6250v1

Mathematics

Number Theory

Scientific paper

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 we do not have $N\in \{ 3,4,8\}$ and also do not have $4\nmid N$ and that all odd prime divisors of $N$ are congruent to $1$ 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.

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