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On Conjugacy Classes of Congruence Subgroups of PSL(2, R)

Published online by Cambridge University Press:  01 February 2010

C. J. Cummins
C. J. Cummins
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada, [email protected]

Abstract

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Let G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup /overline Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 264 - 274
Copyright
Copyright © London Mathematical Society 2009

References

1.Bosma, Wieb, Cannon, John and Playoust, Catherine, ‘The Magma algebra system. I. The user language.’ J. Symbolic Comput. 24 (1997) 235–265. Computational algebra and number theory (London, 1993).Google Scholar
2.Conway, J. H., ‘Understanding groups like Γ0(N).’ ‘Groups, difference sets, and the Monster (Columbus, OH, 1993),’ (de Gruyter, Berlin, 1996), vol. 4 of Ohio State Univ. Math. Res. Inst. Publ. pp. 327–343.Google Scholar
3.Cox, David A. and Parry, Walter R., ‘Genera of congruence subgroups in Q-quaternion algebras.’ J. Reine Angew. Math. 351 (1984) 66–112.Google Scholar
4.Cox, David A. and Parry, Walter R., ‘Genera of congruence subgroups in Q-quaternion algebras.’ Unabridge preprint.Google Scholar
5.Cummins, C. J., ‘Congruence subgroups of groups commensurable with PSL(2; Z) of genus 0 and 1.’ Experiment. Math. 13 (2004) 361–382.Google Scholar
6.Cummins, C. J., ‘fundamental domains for congruence subgroups of genus 0 and 1, ’ submitted for publication.Google Scholar
7.Helling, Heinz, ‘Bestimmung der Kommensurabilitäatsklasse der Hilbertschen Modulgruppe.’ Math. Z. 92 (1966) 269–280.CrossRefGoogle Scholar
8.Sebbar, Abdellah, ‘Classification of torsion-free genus zero congruence groups.’ Proc. Amer. Math. Soc. 129 (2001) 2517–2527(electronic).Google Scholar
9.Thompson, J. G., ‘A finiteness theorem for subgroups of PSL(2; R) which are commensurable with PSL(2; Z).’ ‘The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979),’ (Amer. Math. Soc., Providence, R.I., 1980), vol. 37 of Proc. Sympos. Pure Math. pp. 533–555.Google Scholar
10.Zograf, P., ‘A spectral proof of Rademacher's conjecture for congruence subgroups of the modular group.’ J. Reine Angew. Math. 414 (1991) 113–116.Google Scholar