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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, winfried@mathi.uni-heidelberg.de
Geoffrey Mason
Affiliation:
University of California at Santa Cruz, California 95064, USA, gem@cats.ucsc.edu
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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

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