Nagoya Mathematical Journal

On generalized modular forms and their applications

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 22 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1229955908

Mathematical Reviews number (MathSciNet)
MR2477614

Zentralblatt MATH identifier
1223.11051

Citation

Kohnen, Winfried; Mason, Geoffrey. On generalized modular forms and their applications. Nagoya Math. J. 192 (2008), 119--136. https://projecteuclid.org/euclid.nmj/1229955908

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