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On Generalized Modular forms and their Applications

Published online by Cambridge University Press:  11 January 2016

Winfried Kohnen  and
Geoffrey Mason
Winfried Kohnen
Affiliation:
Mathematisches Institut der Universität, INF 288, D-69120 Heidelberg, Germany, [email protected]
Geoffrey Mason
Affiliation:
University of California at Santa Cruz, California 95064, USA, [email protected]
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Abstract

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We study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

Keywords

11F0311F9917B69
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 192 , 2008 , pp. 119 - 136
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[A] Conway, J. et al., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.Google Scholar
[B] Borcherds, R., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math., 109 (1992), 405–444.Google Scholar
[BKO] Bruinier, J., Kohnen, W. and Ono, K., The arithmetic of the values of modular functions and the divisors of modular forms, Compos. Math., 140 (2004), no. 3, 552–566.Google Scholar
[CN] Conway, J. and Norton, S., Monstrous Moonshine, Bull. Lond. Math. Soc., 12 (1979), 308–339.Google Scholar
[DLM] Dong, C., Li, H., and Mason, G., Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine, Comm. Math. Phys., 214 (2000), 1–56.Google Scholar
[DM1] Dong, C. and Mason, G., Vertex operator algebras and Moonshine: A survey, Adv. Stud. in Pure Math., 24 (1996), 101–136.Google Scholar
[DM2] Dong, C. and Mason, G., Monstrous moonshine of higher weight, Acta Math., 185 (2000), 101–121.CrossRefGoogle Scholar
[ES] Eholzer, W. and Skoruppa, N.-P., Product expansions of conformal characters, Phys. Lett., B 388 (1996), 82–89.Google Scholar
[FHL] Frenkel, I., Huang, Y.-Z., and Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993.Google Scholar
[FLM] Frenkel, I., Lepowsky, J., and Meurman, A., Vertex Operator Algebras and the Monster, Academic Press, San Diego, 1988.Google Scholar
[K] Kohnen, W., On a certain class of modular functions, Proc. Amer. Math. Soc., 133 (2005), no. 1, 65–70.Google Scholar
[KM1] Knopp, M. and Mason, G., Generalized modular forms, J. Number Theory, 99 (2003), 1–18.Google Scholar
[KM2] Knopp, M. and Mason, G., Vector-Valued Modular Forms and Poincaré Series, Ill. J. Math., 48 (2004), no. 4, 1345–1366.Google Scholar
[L] Li, H., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure and Appl. Alg., 96 (1994), 279–297.Google Scholar
[M] Martin, Y., Multiplicative η-quotients, Trans. Amer. Math. Soc., 348 (1996), no. 12, 4825–4856.Google Scholar
[S] Selberg, A., On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math. Vol. VIII, Amer. Math. Soc., Providence R.I., 1965.Google Scholar
[Sh] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Jap. 11, Iwanami Shoten, 1971.Google Scholar
[Z] Zhu, Y., Modular-invariance of characters of vertex operator algebras, J. Amer. Math. Soc., 9 (1996), 237–302.Google Scholar