Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T20:00:08.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Fundamental domains for genus-zero and genus-one congruence subgroups

Part of: Discontinuous groups and automorphic forms Other groups of matrices

Published online by Cambridge University Press:  01 July 2010

C. J. Cummins
C. J. Cummins*
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8, Canada (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines a maximal discrete subgroup Γ0(f)+ of SL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+ is in a subgroup G; and the index of G in Γ0(f)+. The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

MSC classification

Secondary: 20H05: Unimodular groups, congruence subgroups 11F30: Fourier coefficients of automorphic forms 11F22: Relationship to Lie algebras and finite simple groups 20H10: Fuchsian groups and their generalizations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 222 - 245
Copyright
Copyright © London Mathematical Society 2010

References

[1] Beardon, A. F., The geometry of discrete groups, Graduate Texts in Mathematics 91 (Springer, New York, 1983).CrossRefGoogle Scholar
[2] Borcherds, R. E., ‘Monstrous moonshine and monstrous Lie superalgebras’, Invent. Math. 109 (1992) no. 2, 405–444.CrossRefGoogle Scholar
[3] Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265. Computational Algebra and Number Theory (London, 1993).CrossRefGoogle Scholar
[4] Conway, J. H., ‘Understanding groups like Γ0(N)’, Groups, difference sets, and the monster (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications 4 (de Gruyter, Berlin, 1996) 327–343.CrossRefGoogle Scholar
[5] Conway, J. H. and Norton, S. P., ‘Monstrous moonshine’, Bull. London Math. Soc. 11 (1979) no. 3, 308–339.CrossRefGoogle Scholar
[6] Cox, D. A. and Parry, W. R., ‘Genera of congruence subgroups in Q-quaternion algebras’, J. Reine Angew. Math. 351 (1984) 66–112.Google Scholar
[7] Cox, D. A. and Parry, W. R., ‘Genera of congruence subgroups in Q-quaternion algebras’, Preprint (unabridged version of [6]), 1984.CrossRefGoogle Scholar
[8] Cummins, C. J., ‘Congruence subgroups of groups commensurable with PSL(2,ℤ) of genus 0 and 1’, Experiment. Math. 13 (2004) no. 3, 361–382.CrossRefGoogle Scholar
[9] Cummins, C. J., ‘On conjugacy classes of congruence subgroups of PSL(2,ℝ)’, LMS J. Comput. Math. 12 (2009) 264–274.CrossRefGoogle Scholar
[10] Dennin, J. B. Jr, ‘Fields of modular functions of genus 0’, Illinois J. Math. 15 (1971) 442–455.CrossRefGoogle Scholar
[11] Dennin, J. B. Jr, ‘Subfields of K(2n) of genus 0’, Illinois J. Math. 16 (1972) 502–518.CrossRefGoogle Scholar
[12] Dennin, J. B. Jr, ‘The genus of subfields of K(pn)’, Illinois J. Math. 18 (1974) 246–264.CrossRefGoogle Scholar
[13] Ford, L. R., Automorphic functions, 2nd edn (Chelsea Publishing Company, New York, 1951) reprinted by AMS Chelsea Publishing, American Mathematical Society, Providence, RI, 2004.Google Scholar
[14] Ford, D., McKay, J. and Norton, S., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193.CrossRefGoogle Scholar
[15] Frenkel, I., Lepowsky, J. and Meurman, A., Vertex operator algebras and the monster, Pure and Applied Mathematics 134 (Academic Press, Boston, MA, 1988).Google Scholar
[16] Helling, H., ‘Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe’, Math. Z. 92 (1966) 269–280.CrossRefGoogle Scholar
[17] Helling, H., ‘On the commensurability class of the rational modular group’, J. London Math. Soc. (2) 2 (1970) 67–72.CrossRefGoogle Scholar
[18] Jones, G. A., ‘Congruence and noncongruence subgroups of the modular group: a survey’, Proceedings of groups—St Andrews 1985, London Mathematical Society Lecture Note Series 121 (Cambridge University Press, Cambridge, 1986) 223–234.Google Scholar
[19] Knopp, M. I. and Newman, M., ‘Congruence subgroups of positive genus of the modular group’, Illinois J. Math. 9 (1965) 577–583.CrossRefGoogle Scholar
[20] Lehner, J., A short course in automorphic functions (Holt, Rinehart and Winston, New York, 1966).Google Scholar
[21] McKay, J., Unpublished letter to J. Thompson (1978).Google Scholar
[22] McQuillan, D. L., ‘On the genus of fields of elliptic modular functions’, Illinois J. Math. 10 (1966) 479–487.CrossRefGoogle Scholar
[23] McQuillan, D. L., ‘Some results on the linear fractional group’, Illinois J. Math. 10 (1966) 24–38.CrossRefGoogle Scholar
[24] Norton, S. P., ‘More on moonshine’, Computational group theory (Durham, 1982) (Academic Press, London, 1984) 185–193.Google Scholar
[25] Norton, S. P., ‘Non-monstrous moonshine’, Groups, difference sets, and the monster (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications 4 (de Gruyter, Berlin, 1996) 433–441.CrossRefGoogle Scholar
[26] Shimura, G., Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan 11 (Iwanami Shoten, Tokyo, 1971) Kanô Memorial Lectures 1.Google Scholar
[27] Thompson, J. G., ‘Some numerology between the Fischer–Griess Monster and the elliptic modular function’, Bull. London Math. Soc. 11 (1979) no. 3, 352–353.CrossRefGoogle Scholar
[28] 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 (University of California, Santa Cruz, 1979), Proceedings of Symposia in Pure Mathematics 37 (American Mathematical Society, Providence, RI, 1980) 533–555.Google Scholar
[29] 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
Supplementary material: File

Cummins Supplementary Material

Cummins Supplementary Appendix

Download Cummins Supplementary Material(File)
File 17.2 MB