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Lifts of projective congruence groups, II


Author: Ian Kiming
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
Published electronically: July 28, 2014
MathSciNet review: 3251718
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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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Ian Kiming
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
Email: kiming@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9939-2014-12127-7
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.