Fourier coefficients of modular forms of half-integral weight
type: Item
instance of: scholarly article
title: Fourier coefficients of modular forms of half-integral weight
zbMATH Open document ID: 0606.10017
DOI: 10.1007/BF01389423
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143426
review text: The Fourier coefficients \(a_n\) of normalized cusp forms for \(\Gamma_0(4N)\) of half-integral weight \(k\geq 5/2\) are shown to satisfy for \(n\) square-free \[ a_ n=O(n^{k/2-2/7+\varepsilon}). \] This exponent is \(3/14\) more than the Ramanujan-Petersson conjecture would give. This result implies an estimate of values at the center of the critical strip of twisted \(L\)-series related to these cusp forms. The proof uses Petersson's formula for Fourier coefficients of Poincaré series. The resulting sums of Kloosterman sums are estimated by clever averaging. A striking feature is the use of averaging over \(N\). Interesting are also some lemmas on Kloosterman and Salié sums.
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 3979155
zbMATH Keywords: modular L-series
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: normalized cusp forms
zbMATH Keywords: half-integral weight
zbMATH Keywords: sums of Kloosterman sums
zbMATH Keywords: Salié sums
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
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author: H. Iwaniec
description: scientific article; zbMATH DE number 3979155