Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2011-10894-3
Published electronically: November 16, 2011
MathSciNet review: 2869098
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F03, 11F99, 17B69

Retrieve articles in all journals with MSC (2000): 11F03, 11F99, 17B69


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

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