Modular equations and the genus zero property of moonshine functions

Inventiones Mathematicae

Volumn 129, Issue 3, 1997, Pages 413-443

  Cummins, C J ^a   Gannon, T ^b,c  

^a Gina Cody School of Engineering and Computer Science   (Canada)

^b MAX PLANCK INSTITUT FÜR MATHEMATIK   (Germany)

^c YORK UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031490956     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002220050167     Document Type: Article

References (23)

1. R.E. Borcherds, Monstrous Moonshine and monstrous Lie superalgebras, Invent. Math. 109, 405-444 (1992).
2. R.E. Borcherds and A. J. E. Ryba, Modular Moonshine II, Duke Math. J. 83, 435-459 (1996).
3. T. Becker and V. Weispfenning, Gröbner Bases, Springer-Verlag, (1993).
4. H. Cohn and J. McKay, Spontaneous generation of modular invariants, Math, of Comp. 65, 1295-1309 (1996).
5. J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11, 308-339 (1979).
6. C.J. Cummins and S.P. Norton, Rational hauptmoduls are replicable, Can. J. Math. 47, 1201-1218 (1995).
7. I. Chen and N. Yui, Singular values of Thompson series, in Groups, Difference sets and the Monster, eds. K.T. Arusu et al, de Gruyter (1995).
8. P.L. Duren, Univalent Functions, Springer-Verlag (1983).
9. I.B. Frenkel, J. Lepowsky and A. Meurman, Vertex operators and the monster, Academic Press: Boston (1988).
10. G.H. Hardy and E.M. Wright An introduction to the theory of numbers, Fifth Edition, Clarendon Press: Oxford, (1979).
11. A. W. Knapp, Elliptic Curves, Princeton University Press, (1992).
12. M. Koike, On replication formula and Hecke operators, Nagoya University, (preprint).
13. D. N. Kozlov, On completely replicable functions and extremal poset theory, M.Sc. thesis, Department of Mathematics, University of Lund, Sweden, (1994).
14. S. Lang, Elliptic functions, 2nd edition, Addison-Wesley: Reading Mass (1987).
15. K. Mahler, On a class of non-linear functional equations connected with modular functions, J. Austral. Math. Soc. 22A, 65-118 (1976).
16. Y. Martin, On modular invariance of completely replicable functions, in Moonshine, the Monster, and related Topics, eds. C Dong and G. Mason, Contemporary Mathematics 193, 263-286, Amer. Math. Soc., Providence, RI, (1996).
17. A. Markushevich, Theory of functions of a complex variable, Vol. III, Chelsea Publishing Company, (1977).
18. G. Mason (with an appendix by S.P. Norton), Finite groups and modular functions, Proc. Symp. Pure Math. 47, Part 1, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 181-210, Amer. Math. Soc., Providence, RI, (1987).
19. V. K. Murty, Introduction to Abelian Varieties, CRM Monograph Series, Amer. Math. Soc., Providence, RI, (1993).
20. S. P. Norton, More on Moonshine, in Computational Group Theory ed. M. D. Atkinson, Academic Press, 185-193 (1984).
21. S. P. Norton, Non-monstrous Moonshine, in Groups, Difference Sets, and the Monster, eds. K. T. Arasu et al, de Gruyter, 433-441 (1996).
22. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, (1971).
23. J.G. Thompson, A finiteness theorem for subgroups of PSL(2, ℝ) which are commensurable with PSL(2, ℤ), Proc. Sym. Pure. Math., 37, Santa Cruz Conference on finite groups, 533-555, Amer. Math. Soc., Providence RI, (1980).