Identifying congruence subgroups of the modular group

Hsu, Tim

doi:10.1090/S0002-9939-96-03496-X

Identifying congruence subgroups of the modular group
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by Tim Hsu PDF

Proc. Amer. Math. Soc. 124 (1996), 1351-1359 Request permission

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

A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1971, pp. 1–25. MR 0337781
H. Behr and J. Mennicke, A presentation of the groups $\textrm {PSL}(2,\,p)$, Canadian J. Math. 20 (1968), 1432–1438. MR 236269, DOI 10.4153/CJM-1968-144-7
F. Rudolf Beyl, The Schur multiplicator of $\textrm {SL}(2,\textbf {Z}/m\textbf {Z})$ and the congruence subgroup property, Math. Z. 191 (1986), no. 1, 23–42. MR 812600, DOI 10.1007/BF01163607
Juliet Britto, On the construction of noncongruence subgroups, Acta Arith. 33 (1977), no. 3, 261–267. MR 453889, DOI 10.4064/aa-33-3-261-267
Shih-Ping Chan, Mong-Lung Lang, Chong-Hai Lim, and Ser Peow Tan, Special polygons for subgroups of the modular group and applications, Internat. J. Math. 4 (1993), no. 1, 11–34. MR 1209958, DOI 10.1142/S0129167X93000030
Benjamin Fine, Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 129, Marcel Dekker, Inc., New York, 1989. MR 1010229
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.
H. Larcher, The cusp amplitudes of the congruence subgroups of the classical modular group, Illinois J. Math. 26 (1982), no. 1, 164–172. MR 638561, DOI 10.1215/ijm/1256046908
Wilhelm Magnus, Noneuclidean tesselations and their groups, Pure and Applied Mathematics, Vol. 61, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0352287
J. Mennicke, On Ihara’s modular group, Invent. Math. 4 (1967), 202–228. MR 225894, DOI 10.1007/BF01425756
M. H. Millington, On cycloidal subgroups of the modular group, Proc. London Math. Soc. (3) 19 (1969), 164–176. MR 248232, DOI 10.1112/plms/s3-19.1.164
M. H. Millington, Subgroups of the classical modular group, J. London Math. Soc. (2) 1 (1969), 351–357. MR 244160, DOI 10.1112/jlms/s2-1.1.351
Klaus Wohlfahrt, An extension of F. Klein’s level concept, Illinois J. Math. 8 (1964), 529–535. MR 167533