On certain generalized modular forms
type: Item
instance of: scholarly article
title: On certain generalized modular forms
zbMATH Open document ID: 1275.11076
author: Winfried Kohnen
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2010-10-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1285679143
review text: This paper is concerned with a characterization of certain generalized modular forms studied by \textit{M.~Knopp} and \textit{G.~Mason} [J. Number Theory 99, No. 1, 1--28 (2003; Zbl 1074.11025)]. Let \(f\) be a generalized modular form for \(\Gamma_0 (N)\) of weight zero, which has an infinite product expansion of the form \[ f(z) = q^h \prod^\infty_{n=0} (1 - q^n)^{c(n)} \] with \(q= e^{2\pi i z}\). The author proves that div\((f) \neq \emptyset\) if and only if \[ c (n) \ll_\varepsilon n^{- \frac12 +\varepsilon} \] with \(\varepsilon >0\). This result implies that the exponents \(c(n)\) take infinitely many distinct values for such a modular form with \(\mathrm{div}(f) \neq \emptyset\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 5798752
zbMATH Keywords: Generalized modular forms
zbMATH Keywords: divisors
zbMATH Keywords: \(q\)-product expansions
MaRDI profile type: Publication
OpenAlex ID: W1965920237
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cites work: Q4693061
DOI: 10.7169/FACM/1285679143
description: scientific article; zbMATH DE number 5798752