• DOI:10.1090/S0002-9939-2011-10894-3
• Corpus ID: 109926778

On the canonical decomposition of generalized modular functions

@inproceedings{Kohnen2010OnTC,
  title={On the canonical decomposition of generalized modular functions},
  author={Winfried Kohnen and Geoffrey Mason},
  year={2010}
}

• W. Kohnen, G. Mason
• Published 11 March 2010
• Mathematics

The authors have conjectured (\cite{KoM}) 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 (loc cit) in case the divisor of $f$ is \emph{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 \emph{unitary character} and $f_0$ has \emph{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 if the divisor of $f$ is concentrated at the cusps of $\Gamma$.