On generalized modular forms and their applications. (English) Zbl 1223.11051

Kohnen, Winfried; Mason, Geoffrey

Nagoya Math. J. 192, 119-136 (2008).

Summary: We study the Fourier coefficients of generalized modular forms \(f(\tau)\) of integral weight \(k\) on subgroups \(\Gamma\) of finite index in the modular group. We establish two Theorems asserting that \(f(\tau)\) is constant if \(k = 0, f(\tau)\) 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 \(f(\tau)\) has a cuspidal divisor, \(k\) is arbitrary, and \(\Gamma = \Gamma_{0}(N)\), where we show that \(f(\tau)\) 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.

MSC:

11F11	Holomorphic modular forms of integral weight
11F30	Fourier coefficients of automorphic forms
11F22	Relationship to Lie algebras and finite simple groups
17B69	Vertex operators; vertex operator algebras and related structures

References:

[1]	J. Conway et al., Atlas of Finite Groups, Clarendon Press, Oxford, 1985. · Zbl 0568.20001
[2]	R. Borcherds, Monstrous moonshine and monstrous Lie superalgebras , Invent. Math., 109 (1992), 405–444. · Zbl 0799.17014 · doi:10.1007/BF01232032
[3]	J. Bruinier, W. Kohnen and K. Ono, The arithmetic of the values of modular functions and the divisors of modular forms , Compos. Math., 140 (2004), no. 3, 552–566. · Zbl 1060.11019 · doi:10.1112/S0010437X03000721
[4]	J. Conway and S. Norton, Monstrous Moonshine , Bull. Lond. Math. Soc., 12 (1979), 308–339. · Zbl 0424.20010 · doi:10.1112/blms/11.3.308
[5]	C. Dong, H. Li, and G. Mason, Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine , Comm. Math. Phys., 214 (2000), 1–56. · Zbl 1061.17025 · doi:10.1007/s002200000242
[6]	C. Dong and G. Mason, Vertex operator algebras and Moonshine: A survey, Adv. Stud. in Pure Math., 24 (1996), 101–136. · Zbl 0861.17018
[7]	C. Dong and G. Mason, Monstrous moonshine of higher weight , Acta Math., 185 (2000), 101–121. · Zbl 0985.17017 · doi:10.1007/BF02392713
[8]	W. Eholzer and N.-P. Skoruppa, Product expansions of conformal characters , Phys. Lett., B 388 (1996), 82–89. · doi:10.1016/0370-2693(96)01154-9
[9]	I. Frenkel, Y.-Z. Huang, and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993. · Zbl 0789.17022
[10]	I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, San Diego, 1988. · Zbl 0674.17001
[11]	W. Kohnen, On a certain class of modular functions , Proc. Amer. Math. Soc., 133 (2005), no. 1, 65–70. · Zbl 1105.11009 · doi:10.1090/S0002-9939-04-07450-7
[12]	M. Knopp and G. Mason, Generalized modular forms , J. Number Theory, 99 (2003), 1–18. · Zbl 1074.11025 · doi:10.1016/S0022-314X(02)00065-3
[13]	M. Knopp and G. Mason, Vector-Valued Modular Forms and Poincaré Series , Ill. J. Math., 48 (2004), no. 4, 1345–1366. · Zbl 1145.11309
[14]	H. Li, Symmetric invariant bilinear forms on vertex operator algebras , J. Pure and Appl. Alg., 96 (1994), 279–297. · Zbl 0813.17020 · doi:10.1016/0022-4049(94)90104-X
[15]	Y. Martin, Multiplicative \(\eta\)-quotients , Trans. Amer. Math. Soc., 348 (1996), no. 12, 4825–4856. JSTOR: · Zbl 0872.11026 · doi:10.1090/S0002-9947-96-01743-6
[16]	A. Selberg, On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math. Vol. VIII, Amer. Math. Soc., Providence R.I., 1965. · Zbl 0142.33903
[17]	G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Jap. 11, Iwanami Shoten, 1971. · Zbl 0221.10029
[18]	Y. Zhu, Modular-invariance of characters of vertex operator algebras , J. Amer. Math. Soc., 9 (1996), 237–302. JSTOR: · Zbl 0854.17034 · doi:10.1090/S0894-0347-96-00182-8