Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-13T23:43:46.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular Subgroups, Forms, Curves and Surfaces

Published online by Cambridge University Press:  20 November 2018

Abdellah Sebbar
Abdellah Sebbar*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5, e-mail: sebbar@mathstat.uottawa.ca
Rights & Permissions [Opens in a new window]

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.

We study a class of subgroups of $\text{PS}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

Keywords

20H05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 2 , 01 June 2002 , pp. 294 - 308
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Bannai, E., An observation on modular forms. preprint.Google Scholar
[2] Beauville, A., Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières. C. R. Acad. Sci. Paris Sér. I Math. (19) 294 (1982), 294–1982.Google Scholar
[3] Conway, J. H., Understanding groups like Γ0(N). Groups, difference sets, and the Monster, Columbus, Ohio, 1993, 327–343, Ohio State Univ. Math. Res. Inst. Publ. 4, de Gruyter, Berlin, 1996.Google Scholar
[4] Conway, J. H. and Norton, S. P., Monstrous moonshine. Bull. London Math. Soc. (3) 11 (1979), 11–1979.Google Scholar
[5] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups. Grundlehren der Mathematischen Wissenschaften Springer-Verlag, New York, 1993.Google Scholar
[6] Conway, J. H. and Sloane, N. J. A., Low-Dimensional Lattices II: Subgroups of GL(n, Z). Proc. Roy. Soc. London Ser. A 419 (1988), 419–1988.Google Scholar
[7] Ford, D., McKay, J. and Norton, S., More on replicable functions. Comm. Algebra 22 (1994), 22–1994.Google Scholar
[8] Kodaira, K., On compact complex analytic surfaces II. Ann. of Math. 77 (1963), 77–1963.Google Scholar
[9] McKay, J. and Sebbar, A., Fuchsian groups, automorphic functions and Schwarzians. Math. Ann. (2) 318 (2000), 318–2000.Google Scholar
[10] McKay, J. and Sebbar, A., J—invariants of the arithmetic semistable elliptic surfaces and graphs. AMS-CRM lecture notes series, to appear.Google Scholar
[11] Millington, M. H., Subgroups of the classical modular group. J. London Math. Soc. (2) 1 (1969), 1–1969.Google Scholar
[12] Miranda, R. and Persson, U., On extremal rational elliptic surfaces. Math. Z. (4) 193 (1986), 193–1986.Google Scholar
[13] Rains, E. M. and Sloane, N. J. A., The shadow theory of modular and unimodular lattices. J. Number Theory (2) 73 (1998), 73–1998.Google Scholar
[14] Rankin, Robert A., Modular forms and functions. Cambridge University Press, 1977.Google Scholar
[15] Sebbar, A., Classification of genus zero torsion-free congruence groups in PSL2(Z). Proceedings of the Amer. Math. Soc., to appear.Google Scholar
[16] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, 1971.Google Scholar
[17] Shioda, T., On elliptic modular surfaces. J.Math. Soc. Japan 24 (1972), 24–1972.Google Scholar