Lifting modular forms of half-integral weight to Siegel modular forms of even genus
type: Item
instance of: scholarly article
title: Lifting modular forms of half-integral weight to Siegel modular forms of even genus
zbMATH Open document ID: 1004.11020
author: Winfried Kohnen
published in: Mathematische Annalen
publication date: 2002-06-10T00:00:00Z
review text: Let \(f\) be a normalized Hecke eigenform in the space \(S_{2k}\) of cusp forms of even weight \(2k\) on the full modular group \(\text{SL}_2(\mathbb{Z})\), and let \(n\) be a positive integer for which \(n-k\) is even. Then there exists a Hecke eigenform \(F\) in the space \(S_{k+n}(\Gamma_{2n})\) of cusp forms of weight \(k+n\) with respect to the Siegel modular group \(\Gamma_{2n}= \text{Sp}_{2n}(\mathbb{Z})\) of even genus \(2n\), such that the standard zeta function of \(F\) is \[ \zeta(s) \prod_{j=1}^{2n} L(f,s+k+n-j), \] where \(L(f,s)\) is the Hecke \(L\)-function of \(f\). This was conjectured by W. Duke and Ö. Imamoglu, and it was proved by \textit{T. Ikeda} [Ann. Math. (2) 154, 641-681 (2001; Zbl 0998.11023)]. Ikeda also proved a formula for the coefficients \(A(T)\) in the Fourier expansion of \(F(Z)\). One of the factors in \(A(T)\) is a coefficient of the cusp form \(g\) of half-integral weight \(k+ \frac 12\) in the Kohnen plus space \(S_{k+1/2}^+\) which corresponds to \(f\) via the Shimura lift. In the paper under review, another formula for \(A(T)\) is proved. It also contains coefficients of \(g\), but its other ingredients are easier to understand than those in Ikeda's formula. The author uses his formula to define a linear map from \(S_{k+1/2}^+\) to \(S_{k+n}(\Gamma_{2n})\) which on Hecke eigenforms coincides with Ikeda's lifting map. Many interesting details of the paper cannot be reproduced here. It ends with a discussion of some open problems. One of them is a conjecture on the image of the lifting map in \(S_{k+n} (\Gamma_{2n})\) which for \(n=1\) is equal to the Maass space.
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 1753094
zbMATH Keywords: normalized Hecke eigenform
zbMATH Keywords: Hecke eigenform
zbMATH Keywords: Siegel modular group
zbMATH Keywords: standard zeta function
zbMATH Keywords: half-integral weight
zbMATH Keywords: Ikeda's lifting map
MaRDI profile type: Publication
full work available at URL: s002080100285
OpenAlex ID: W2082995338
DOI: 10.1007/S002080100285
description: scientific article; zbMATH DE number 1753094