Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T22:46:11.668Z Has data issue: false hasContentIssue false
  • English
  • Français

On Functions Satisfying Modular Equations for Infinitely Many Primes

Published online by Cambridge University Press:  20 November 2018

Dmitry N. Kozlov
Dmitry N. Kozlov*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email: kozlov@math.ias.edu, kozlov@math.mit.edu, kozlov@math.kth.se
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.

In this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are:

  1. 1) every such function is analytic in the upper half plane;

  2. 2) if such function takes the same value in two different points ${{z}_{1}}$ and ${{z}_{2}}$ then there exists an $f$-preserving analytic bijection between neighbourhoods of ${{z}_{1}}$ and ${{z}_{2}}$.

Keywords

11Mxx
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 5 , 01 October 1999 , pp. 1020 - 1034
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Alexander, D., Cummins, C., McKay, J. and Simons, C., Completely replicable functions. In: Groups, combinatorics and geometry (Durham, 1990), Cambridge Univ. Press, Cambridge, 1992, 87–98.Google Scholar
[2] Borcherds, R. E., Monstrous moonshine and monstrous Lie superalgebras. Invent.Math. 109(1992), 405–444.Google Scholar
[3] Borcherds, R. E. and Ryba, A. J. E., Modular Moonshine II. Duke Math. J. 83(1996), 435–459.Google Scholar
[4] Cohn, H. and McKay, J., Spontaneous generation of modular invariants. Math. Comp. 65(1996), 1295–1309.Google Scholar
[5] Cohn, H. and McKay, J., Modular functions from nothing. Preprint, 1994.Google Scholar
[6] Conway, J. H. and Norton, S. P., Monstrous moonshine. Bull. LondonMath. Soc. 11(1979), 308–339.Google Scholar
[7] Cummins, C. J. and Gannon, T., Modular equations and the genus zero property of moonshine functions. Invent. Math. (3) 129(1998), 413–443.Google Scholar
[8] Frenkel, I. B., Lepowsky, J. and Meurman, A., Vertex operators and theMonster. Academic Press, Boston, 1988.Google Scholar
[9] Mahler, K., On a class of non-linear functional equations connected with modular functions. J. Austral. Math. Soc. 22(A)(1976), 65–118.Google Scholar
[10] Martin, Y., On modular invariance of completely replicable functions. In: Moonshine, theMonster, and related topics (South Hadley, MA, 1994), Contemp.Math. 193(1996), 263–286.Google Scholar
[11] McKay, J. and Strauss, H., The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18(1990), 253–278.Google Scholar
[12] Norton, S. P., More on moonshine. In: Computational Group Theory (ed. Atkinson, M. D.), Academic Press, 1984, 185–193.Google Scholar
[13] Norton, S. P., Non-monstrous Moonshine. In: “Groups, Difference Sets, and theMonster” (eds. Arasu, K. T. et al.), de Gruyter, 1996, 433–441.Google Scholar