Multiplicative 𝜂-quotients

Martin, Yves

doi:10.1090/S0002-9947-96-01743-6

Multiplicative $\eta$-quotients
HTML articles powered by AMS MathViewer

by Yves Martin PDF

Trans. Amer. Math. Soc. 348 (1996), 4825-4856 Request permission

Abstract:

Let $\eta (z)$ be the Dedekind $\eta$-function. In this work we exhibit all modular forms of integral weight $f(z) = \eta (t_1z)^{r_1}\eta (t_2z)^{r_2}\dots \eta (t_sz)^{r_s}$, for positive integers $s$ and $t_j$ and arbitrary integers $r_j$, such that both $f(z)$ and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group $2 \mathrm {Co}_1$ via a generalized McKay-Thompson series.

References

A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
Anthony J. F. Biagioli, Eta-products which are simultaneous eigenforms of Hecke operators, Glasgow Math. J. 35 (1993), no. 3, 307–323. MR 1240373, DOI 10.1017/S0017089500009885
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI 10.1112/blms/11.3.308
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
D. Dummit, H. Kisilevsky, and J. McKay, Multiplicative products of $\eta$-functions, Finite groups—coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 89–98. MR 822235, DOI 10.1090/conm/045/822235
Basil Gordon and Dale Sinor, Multiplicative properties of $\eta$-products, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp. 173–200. MR 1019331, DOI 10.1007/BFb0086404
K. Harada, Moonshine of finite groups, Lecture Notes of The Ohio State University.
Marvin I. Knopp, Modular functions in analytic number theory, Markham Publishing Co., Chicago, Ill., 1970. MR 0265287
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
Masao Koike, On McKay’s conjecture, Nagoya Math. J. 95 (1984), 85–89. MR 759465, DOI 10.1017/S0027763000020985
Masao Koike, Mathieu group $M_{24}$ and modular forms, Nagoya Math. J. 99 (1985), 147–157. MR 805086, DOI 10.1017/S0027763000021541
Takeshi Kondo, Examples of multiplicative $\eta$-products, Sci. Papers College Arts Sci. Univ. Tokyo 35 (1986), no. 2, 133–149. MR 847884
Mong-Lung Lang, On a question raised by Conway-Norton, J. Math. Soc. Japan 41 (1989), no. 2, 263–284. MR 984751, DOI 10.2969/jmsj/04120263
Wen Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315. MR 369263, DOI 10.1007/BF01344466
Y. Martin, On multiplicative $\eta$-quotients, PhD. Thesis, University of California, Santa Cruz (1993).
Geoffrey Mason, Frame-shapes and rational characters of finite groups, J. Algebra 89 (1984), no. 2, 237–246. MR 751142, DOI 10.1016/0021-8693(84)90215-1
Geoffrey Mason, $M_{24}$ and certain automorphic forms, Finite groups—coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 223–244. MR 822240, DOI 10.1090/conm/045/822240
Geoffrey Mason, Finite groups and modular functions, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 181–210. With an appendix by S. P. Norton. MR 933359
Geoffrey Mason, Finite groups and Hecke operators, Math. Ann. 283 (1989), no. 3, 381–409. MR 985239, DOI 10.1007/BF01442736
Geoffrey Mason, Elliptic systems and the eta-function, Notas Soc. Mat. Chile 8 (1989), no. 1, 37–53. MR 1051836
Geoffrey Mason, On a system of elliptic modular forms attached to the large Mathieu group, Nagoya Math. J. 118 (1990), 177–193. MR 1060709, DOI 10.1017/S0027763000003068
J. G. Thompson, Finite groups and modular functions, Bull. London Math. Soc. 11 (1979), no. 3, 347–351. MR 554401, DOI 10.1112/blms/11.3.347