Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T02:37:27.610Z Has data issue: false hasContentIssue false
  • English
  • Français

The Schwarzian equation for completely replicable functions

Part of: Representation theory of groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 October 2017

Abdelkrim El Basraoui  and
John McKay
Abdelkrim El Basraoui
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N6N5, Canada email [email protected]
John McKay
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.

We describe the Schwarzian equations for the 328 completely replicable functions with integral $q$-coefficients [Ford et al., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193].

MSC classification

Primary: 11F03: Modular and automorphic functions 11F22: Relationship to Lie algebras and finite simple groups
Secondary: 20D08: Simple groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 20 , Issue 1 , 2017 , pp. 30 - 52
Copyright
© The Author(s) 2017 

References

Alexander, D., Cummins, C., McKay, J. and Simons, C., ‘Completely replicable functions’, Groups, combinatorics & geometry (Durham, 1990), London Mathematical Society Lecture Note Series 165 (Cambridge University Press, Cambridge, 1992) 87–98.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray (Oxford University Press, Eynsham, 1985).Google Scholar
Conway, J. H. and Norton, S. P., ‘Monstrous moonshine’, Bull. Lond. Math. Soc. 11 (1979) no. 3, 308–339.Google Scholar
Dedekind, R., ‘Über Die Elliptischen Modul-Functionen’, J. reine angew. Math. 3 (1877) no. 2, 265–292.Google Scholar
El Basraoui, A. and Sebbar, A., ‘Rational equivariant forms’, Int. J. Number Theory 8 (2012) no. 4, 963–981.Google Scholar
Ford, D., Automorphic functions (Princeton University Press, 1951).Google Scholar
Ford, D., McKay, J. and Norton, S., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193.Google Scholar
Harnard, J. and McKay, J., ‘Modular solutions to equations of generalized Halphen type’, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 456 (2000) no.1994, 261–294.Google Scholar
Klein, F., The icosahedron and the solution of equations of the fifth degree (Nauka, Moscow, 1989). Translated from the German by A. L. Gorodentsev and A. A. Kirillov. Translation edited and with a preface by A. N. Tyurin. With appendices by V. I. Arnold, J.-P. Serre and A. N. Tyurin.Google Scholar
McKay, J. and Sebbar, A., ‘Fuchsian groups, automorphic functions and Schwarzians’, Math. Ann. 318 (2000) no. 2, 255–275.Google Scholar
Ovsienko, V. and Tabachnikov, S., Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups , Cambridge Tracts in Mathematics 165 (Cambridge University Press, Cambridge, 2005).Google Scholar
Poincaré, H., ‘Mémoire sur les fonctions fuchsiennes’, Acta Math. 1 (1882) no. 1, 193–294 (French).Google Scholar
Rankin, R. A., Modular forms and functions (Cambridge University Press, 1977).Google Scholar
Serre, J.-P., A course in arithmetic , Graduate Texts in Mathematics 7 (Springer, New York–Heidelberg, 1973). Translated from the French.Google Scholar