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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the canonical decomposition of generalized modular functions


Authors: Winfried Kohnen and Geoffrey Mason
Journal: Proc. Amer. Math. Soc. 140 (2012), 1125-1132
MSC (2000): Primary 11F03, 11F99, 17B69
Published electronically: November 16, 2011
MathSciNet review: 2869098
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Abstract: The authors have conjectured that if a normalized generalized modular function (GMF) $ f$, defined on a congruence subgroup $ \Gamma $, has integral Fourier coefficients, then $ f$ is classical in the sense that some power $ f^m$ is a modular function on $ \Gamma $. A strengthened form of this conjecture was proved in case the divisor of $ f$ is empty. In the present paper we study the canonical decomposition of a normalized parabolic GMF $ f = f_1f_0$ into a product of normalized parabolic GMFs $ f_1, f_0$ such that $ f_1$ has unitary character and $ f_0$ has empty divisor. We show that the strengthened form of the conjecture holds if the first ``few'' Fourier coefficients of $ f_1$ are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either $ f_0=1$ or the divisor of $ f$ is concentrated at the cusps of $ \Gamma $.


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Additional Information

Winfried Kohnen
Affiliation: Mathematisches Institut der Universität Heidelberg, INF 288, D-69120 Heidelberg, Germany
Email: winfried@mathi.uni-heidelberg.de

Geoffrey Mason
Affiliation: Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064
Email: gem@cats.ucsc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10894-3
PII: S 0002-9939(2011)10894-3
Keywords: Canonical decomposition, generalized modular function
Received by editor(s): August 12, 2010
Received by editor(s) in revised form: November 9, 2010
Published electronically: November 16, 2011
Additional Notes: The second author was supported in part by the NSF and NSA
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society