On a theorem of Vignéras
type: Item
instance of: scholarly article
title: On a theorem of Vignéras
zbMATH Open document ID: 0954.11016
DOI: 10.1007/BF02942560
author: Jan Hendrik Bruinier
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: Let \(f(z)=\sum_{n\geq 0}a(n) e^{2\pi inz}\) be a non-zero modular form of half-integral weight \(k+{1\over 2}\) on \(\Gamma_1(N)\). The author proves several criteria for the nonvanishing of the coefficients \(a(n)\). One of them is: If \(k\geq 2\), or if \(k=1\) and \(f\) is a cusp form in the orthogonal complement of the space spanned by theta series in one variable, then for any prime \(p\nmid N\) and any \(\varepsilon\in\{1,-1\}\) there is some \(n\) such that \(({n\over p})=\varepsilon\) and \(a(n)\neq 0\). The results imply an earlier theorem of \textit{M. F. Vignéras} on the nonvanishing of \(a(n)\) [Lect. Notes Math. 627, 79-103 (1977; Zbl 0373.10027)]. The proof uses well-known operators on modular forms and the theorem of \textit{J.-P. Serre} and \textit{H. M. Stark} on the generation of modular forms of weight \({1\over 2}\) by theta series [Lect. Notes Math. 627, 27-67 (1977; Zbl 0371.10019)].
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 1416673
zbMATH Keywords: modular form of half-integral weight
zbMATH Keywords: nonvanishing of the coefficients
MaRDI profile type: Publication
cites work: Nonvanishing theorems for L-functions of modular forms and their derivatives
cites work: On the Niwa-Shintani theta-kernel lifting of modular forms
cites work: Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
cites work: A simple remark on eigenvalues of Hecke operators on Siegel modular forms
cites work: Fourier coefficients and modular forms of half-integral weight
cites work: Values of L-series of modular forms at the center of the critical strip
cites work: Mean values of derivatives of modular \(L\)-series
cites work: Modular forms of half integral weight and the integral of certain theta-functions
cites work: Q4146791
cites work: On modular forms of half integral weight
cites work: Theta series of weight 3/2
cites work: Q4149531
cites work: Sur les coefficients de Fourier des formes modulaires de poids demi-entier
description: scientific article; zbMATH DE number 1416673