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) , defined on a congruence subgroup , has integral Fourier coefficients, then is classical in the sense that some power is a modular function on . A strengthened form of this conjecture was proved in case the divisor of is *empty*. In the present paper we study the canonical decomposition of a normalized parabolic GMF into a product of normalized parabolic GMFs such that has *unitary character* and has *empty divisor*. We show that the strengthened form of the conjecture holds if the first ``few'' Fourier coefficients of are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either or the divisor of is concentrated at the cusps of .

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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

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