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Rational Hauptmoduls are Replicable

Published online by Cambridge University Press:  20 November 2018

C. J. Cummins  and
S. P. Norton
C. J. Cummins
Affiliation:
Centre Interuniversitaire en Calcul Mathématique Algébrique Department of Mathematics and Statistics Concordia University 1455 de Maisonneuve Boulevard West Montréal, Québec H3G 1M8
S. P. Norton
Affiliation:
Department of Pure Mathematics and Mathematical Statistics Cambridge University16 Mill Lane Cambridge, CB2 1SB England
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Abstract

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It is shown that if ƒ is a Hauptmodul with rational integer coefficients for a group G < PGL2(ℚ)+, of genus zero, containing a with finite index and z ⟼ z+k precisely when k is an integer, then ƒ is replicable. Examples of such functions are given by the Moonshine functions described by Conway and Norton [CN].

Keywords

11F0311F2230F35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1201 - 1218
Copyright
Copyright © Canadian Mathematical Society 1995

References

[ACMS] Alexander, D., Cummins, C., McKay, J. and Simons, C., Completely replicable functions. In: Groups, Combinatorics and Geometry, Lecture Notes in Math., (ed. Liebeck, M.W. and Saxl, J.), Cambridge Univ. Press, 1992. 87-95Google Scholar
[ATLAS] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A., Atlas of finite groups, Oxford Univ. Press, 1985.Google Scholar
[B1] Borcherds, R.E., Monstrous Moonshine and monstrous Lie super algebras, Invent. Math. 109(1992), 405–444.Google Scholar
[CN] Conway, J.H. and Norton, S.P., Monstrous Moonshine, Bull. London Math. Soc. 11(1979), 308–339.Google Scholar
[F] Ferenbaugh, C.R., On the Modular Functions involved in “Monstrous Moonshine ”, Ph.D. thesis, Princeton University, 1992.Google Scholar
[K] Koike, M., On replication formula and Hecke operators, Nagoya University, preprint.Google Scholar
[N1] Norton, S.P., More on Moonshine. In: Computational Group Theory (éd. Atkinson, M.D.), Academic Press, 1984. 185–193.Google Scholar
[N2] Norton, S.P., Non-monstrous Moonshine, Proceedings of the Columbus conference on the Monster, 1993. to appear.Google Scholar
[Sh] Shimura, G. Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971.Google Scholar
[Sm] Smith, G.W., Higher genus Moonshine, 1993. preprint.Google Scholar
[T] Thompson, J.G., Some numerology between the Fischer-Griess monster and the elliptic modular function,, Bull. London Math. Soc. 11(1979), 352–353.Google Scholar