Lifts of projective congruence groups, II
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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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Additional Information
- Ian Kiming
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
- Email: kiming@math.ku.dk
- Received by editor(s): August 10, 2012
- Received by editor(s) in revised form: December 21, 2012
- Published electronically: July 28, 2014
- Communicated by: Matthew A. Papanikolas
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3761-3770
- MSC (2010): Primary 11F06; Secondary 20H05
- DOI: https://doi.org/10.1090/S0002-9939-2014-12127-7
- MathSciNet review: 3251718