Nagoya Mathematical Journal

On generalized modular forms and their applications

Winfried Kohnen and Geoffrey Mason

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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

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

First available in Project Euclid: 22 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F03: Modular and automorphic functions 11F99: None of the above, but in this section 17B69: Vertex operators; vertex operator algebras and related structures


Kohnen, Winfried; Mason, Geoffrey. On generalized modular forms and their applications. Nagoya Math. J. 192 (2008), 119--136.

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