On the growth of Fourier coefficients of certain special Siegel cusp forms
type: Item
instance of: scholarly article
title: On the growth of Fourier coefficients of certain special Siegel cusp forms
zbMATH Open document ID: 1072.11032
author: Winfried Kohnen
published in: Mathematische Zeitschrift
publication date: 2005-02-01T00:00:00Z
review text: Let \(S_k(\Gamma_n)\) denote the space of Siegel cusp forms of weight \(k\) and genus \(n\). A conjecture of \textit{H.~L. Resnikoff} and \textit{R. L. Saldaña} [J. Reine Angew. Math. 265, 90--109 (1974; Zbl 0278.10028)] claims that the Fourier coefficients \(A(T)\) of any \(F\in S_k(\Gamma_n)\) satisfy \[ A(T)\ll_{\varepsilon,F}(\text{det\,}T)^{{k\over 2}-{n+1\over 5}+\varepsilon}\qquad (\varepsilon> 0)\tag{1} \] for all half-integral symmetric matrices \(T> 0\) of size \(n\). This conjecture is true for \(n= 1\) by Deligne's theorem (previously the Ramanujan-Petersson conjecture) whereas for \(n> 1\) counterexamples are known. The author adds to the list of counterexamples and proves: Let \(n\equiv 1\pmod 4\), \(n\equiv k\mod 2\) and let \(F\in S_{k+n}(\Gamma_{2n})\) be a Hecke eigenform that is an Ikeda lift of a normalized Hecke eigenform \(f\in S_{2k}(\Gamma_1)\). Then estimate (1) does not hold true.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 2131982
zbMATH Keywords: Siegel cusp form
zbMATH Keywords: Fourier coefficient
zbMATH Keywords: Ikeda lift
MaRDI profile type: Publication
full work available at URL: s00209-003-0643-7
OpenAlex ID: W2070387982
DOI: 10.1007/S00209-003-0643-7
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description: scientific article; zbMATH DE number 2131982