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Identifying congruence subgroups
of the modular group


Author: Tim Hsu
Journal: Proc. Amer. Math. Soc. 124 (1996), 1351-1359
MSC (1991): Primary 20H05; Secondary 20F05
DOI: https://doi.org/10.1090/S0002-9939-96-03496-X
MathSciNet review: 1343700
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Abstract | References | Similar Articles | Additional Information

Abstract: We exhibit a simple test (Theorem 2.4) for determining if a given (classical) modular subgroup is a congruence subgroup, and give a detailed description of its implementation (Theorem 3.1). In an appendix, we also describe a more ``invariant'' and arithmetic congruence test.


References [Enhancements On Off] (What's this?)

  • 1. A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Proc. Symp. Pure Math., Combinatorics (Providence) (T. S. Motzkin, ed.), vol. 19, AMS, Providence, 1971, pp. 1--26. MR 49:2550
  • 2. H. Behr and J. Mennicke, A presentation of the groups ${PSL}(2,p)$, Can. J. Math. 20 (1968), 1432--1438. MR 38:4566
  • 3. F. R. Beyl, The Schur multiplicator of $SL(2,Z/mZ)$ and the congruence subgroup property, Math. Z. 191 (1986), 23--42. MR 87b:20071
  • 4. J. Britto, On the construction of non-congruence subgroups, Acta Arith. XXXIII (1977), 261--267. MR 56:12142
  • 5. S.-P. Chan, M.-L. Lang, C.-H. Lim, and S.-P. Tan, Special polygons for subgroups of the modular group and applications, Internat. J. Math. 4 (1993), no. 1, 11--34. MR 94j:11045
  • 6. B. Fine, Algebraic theory of the Bianchi groups, Marcel Dekker, Inc., New York, 1989. MR 90h:20002
  • 7. M.-L. Lang, C.-H. Lim, and S.-P. Tan, An algorithm for determining if a subgroup of the modular group is congruence, preprint, 1992.
  • 8. H. Larcher, The cusp amplitudes of the congruence subgroups of the classical modular group, Ill. J. Math. 26 (1982), no. 1, 164--172. MR 83a:10040
  • 9. W. Magnus, Noneuclidean tesselations and their groups, Academic Press, 1974. MR 50:4774
  • 10. J. Mennicke, On Ihara's modular group, Invent. Math. 4 (1967), 202--228. MR 37:1485
  • 11. M. H. Millington, On cycloidal subgroups of the modular group, Proc. Lon. Math. Soc. 19 (1969), 164--176. MR 40:1484
  • 12. ------, Subgroups of the classical modular group, J. Lon. Math. Soc. 1 (1969), 351--357. MR 39:5477
  • 13. K. Wohlfahrt, An extension of F. Klein's level concept, Ill. J. Math. 8 (1964), 529--535. MR 29:4805

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Additional Information

Tim Hsu
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: timhsu@math.princeton.edu, timhsu@math.lsa.umich.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03496-X
Keywords: Congruence subgroups, classical modular group
Received by editor(s): September 1, 1994
Additional Notes: The author was supported by an NSF graduate fellowship and DOE GAANN grant #P200A10022.A03.
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1996 American Mathematical Society

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