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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the canonical decomposition of generalized modular functions
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by Winfried Kohnen and Geoffrey Mason PDF
Proc. Amer. Math. Soc. 140 (2012), 1125-1132 Request permission

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 $\Gamma$.
References
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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
  • MR Author ID: 189334
  • Email: gem@cats.ucsc.edu
  • 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
  • © Copyright 2011 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1125-1132
  • MSC (2000): Primary 11F03, 11F99, 17B69
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10894-3
  • MathSciNet review: 2869098