Remark on the dimension of Kohnen's spaces of half-integral weight with square free level
type: Item
instance of: scholarly article
title: Remark on the dimension of Kohnen's spaces of half-integral weight with square free level
zbMATH Open document ID: 1010.11024
author: Hisashi Kojima
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-08-18T00:00:00Z
review text: For an odd integer \(N\), a Dirichlet character \(\chi\) modulo \(N\), and an integer \(k\geq 0\), consider the space \(S_{k+1/2} (4N,\chi_\varepsilon)\) of cusp forms of half-integral weight \(k+{1\over 2}\), level \(4N\) and character \(\chi_\varepsilon\), where \(\varepsilon= \chi(-1)\) and \(\chi_\varepsilon= ({4 \varepsilon\over \cdot})\chi\). Kohnen's subspace \(S_{k+1/2} (N,\chi)\) consists of all those \(f\) in \(S_{k+1/2} (4N,\chi_\varepsilon)\) whose Fourier expansion is of the form \(f(z)= \sum_na(n) e^{2\pi inz}\) where the summation is restricted by the condition that \(\varepsilon(-1)^kn \equiv 0\) or 1 \(\pmod 4\). The author uses a result of \textit{H. Cohn} and \textit{J. Oesterlé} [Lect. Notes Math. 627, 69-78 (1977; Zbl 0371.10020)] on the dimension of \(S_{k+ 1/2}(4N, \chi_\varepsilon)\) to prove that \(\dim S_{k+1/2} (N,\chi)= \dim S_{2k} (N,\chi^2)\) if \(N\) is square-free and \(k\geq 2\). The difference \(\dim S_{k+1/2} (N,\chi)- {1\over 3}\dim S_{k+1/2} (4N,\chi_\varepsilon)\) is expressed in elementary terms.
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 1782993
zbMATH Keywords: cusp forms of half-integral weight
zbMATH Keywords: Kohnen's subspace
zbMATH Keywords: dimension
MaRDI profile type: Publication
full work available at URL: pjaa.78.18
OpenAlex ID: W2005756487
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DOI: 10.3792/PJAA.78.18
description: scientific article; zbMATH DE number 1782993