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.
References
J. Conway et al., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
Mathematical Reviews (MathSciNet):
MR827219
R. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math., 109 (1992), 405--444.
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.
J. Conway and S. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc., 12 (1979), 308--339.
Mathematical Reviews (MathSciNet):
MR554399
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.
C. Dong and G. Mason, Vertex operator algebras and Moonshine: A survey, Adv. Stud. in Pure Math., 24 (1996), 101--136.
C. Dong and G. Mason, Monstrous moonshine of higher weight, Acta Math., 185 (2000), 101--121.
W. Eholzer and N.-P. Skoruppa, Product expansions of conformal characters, Phys. Lett., B 388 (1996), 82--89.
I. Frenkel, Y.-Z. Huang, and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993.
I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, San Diego, 1988.
Mathematical Reviews (MathSciNet):
MR996026
W. Kohnen, On a certain class of modular functions, Proc. Amer. Math. Soc., 133 (2005), no. 1, 65--70.
M. Knopp and G. Mason, Generalized modular forms, J. Number Theory, 99 (2003), 1--18.
M. Knopp and G. Mason, Vector-Valued Modular Forms and Poincaré Series, Ill. J. Math., 48 (2004), no. 4, 1345--1366.
H. Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure and Appl. Alg., 96 (1994), 279--297.
Y. Martin, Multiplicative $\eta$-quotients, Trans. Amer. Math. Soc., 348 (1996), no. 12, 4825--4856.
A. Selberg, On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math. Vol. VIII, Amer. Math. Soc., Providence R.I., 1965.
Mathematical Reviews (MathSciNet):
MR182610
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Jap. 11, Iwanami Shoten, 1971.
Mathematical Reviews (MathSciNet):
MR314766
Y. Zhu, Modular-invariance of characters of vertex operator algebras, J. Amer. Math. Soc., 9 (1996), 237--302.
