On the Shimura correspondence for \(GSp(4)\)
type: Item
instance of: scholarly article
title: On the Shimura correspondence for \(GSp(4)\)
zbMATH Open document ID: 0713.11034
DOI: 10.1007/BF01459242
author: Solomon Friedberg
author: Shek-Tung Wong
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164815
review text: If \(\tilde f\) is a holomorphic modular form of half-integral weight, Shimura showed that one can associate to \(\tilde f\) an automorphic form f of integral weight. Moreover, he showed that the L-series for \(f\) can be expressed as a Rankin-Selberg integral of \(\tilde f\) with a \(GL(2)\) theta series. In this paper an analogue of the Shimura construction for generic automorphic forms of half-integral weight on GSp(4) (\(4\times 4\) matrices) is given. Let \({\tilde \phi}\) be an automorphic form of half-integral weight on \(GSp(4)\). The Shimura correspondent \(\phi\) is the automorphic form on \(Sp(4)\cong O(3,2)\) arising from the dual pair \((\tilde Sp(4), O(3,2))\) (the tilde denotes the metaplectic double cover). The L-group of \(O(3,2)\) is \(Sp(4,{\mathbb{C}})\), which has two standard representations, one of degree 4 and the other of degree 5. Here, for generic \({\tilde \phi}\) the degree 5 L-function \(L(s,\phi,5)\) is expressed as an integral involving \({\tilde \phi}\), and this is used to give its analytic continuation and functional equation. To give this integral representation, a metaplectic version of Novodvorsky's integral for the Rankin-Selberg convolution on \(GSp(4)\times GL(2)\) is used. Feeding a \(GL(2)\) theta series into this construction, one obtains a Dirichlet series summed over (Whittaker)-Fourier coefficients of \({\tilde \phi}\) indexed by squares. To compute this series, the Hecke operators on \(GSp(4)\) are computed. Provided \({\tilde \phi}\) is a Hecke eigenform, the Hecke relations imply that this Dirichlet series has an Euler product, and that at almost all places the local factor is the local component of \(L(s,\phi,5)\). Combining with the explicit functional equation for the necessary Eisenstein series, one obtains the analytic continuation and functional equation. The Shimura method has been used to obtain Euler products with functional equations in other metaplectic contexts, by Gelbart and Piatetski-Shapiro and by Bum…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F60
zbMATH DE Number: 4173087
zbMATH Keywords: metaplectic automorphic form
zbMATH Keywords: Shimura correspondent
zbMATH Keywords: Novodvorsky's integral for the Rankin-Selberg convolution
zbMATH Keywords: theta series
zbMATH Keywords: Hecke operators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4173087