A converse theorem for \(GL_ 4\)
type: Item
instance of: scholarly article
title: A converse theorem for \(GL_ 4\)
zbMATH Open document ID: 0864.22009
published in: Mathematical Research Letters
publication date: 1997-06-22T00:00:00Z
review text: Suppose \(k\) is a global field, \(\mathbb{A}\) the adeles over \(k\). The authors sketch the proofs of two converse theorems for \(GL_4(\mathbb{A})\). The first of these converse theorems can be stated as follows. Assume \(\pi\) is an irreducible admissible generic representation of \(GL_4 (\mathbb{A})\) with a central character trivial on the principal ideles \(k^\times\). Assume that \(\tau\) is an automorphic cuspidal representation of \(GL_m (\mathbb{A})\), for \(m=1,2\). Assume that for every \(\tau\) the function \(L(\pi \times\tau,s)\) is nice, meaning that it and the \(L\)-function for \(\widetilde \pi\times \widetilde\tau\) have analytic continuation to entire functions of \(s\), bounded in any vertical strip and satisfying a functional equation. Then the conclusion of the first converse theorem is that \(\pi\) is a cuspidal automorphic representation of \(GL_4(\mathbb{A})\). The second theorem excludes \(\tau\) unramified at a finite set \(S\) of places. The conclusion is that \(\pi_v\simeq\pi_v'\), for \(v\notin S\), where \(\pi'\) is an automorphic representation of \(GL_4 (\mathbb{A})\).
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 913096
zbMATH Keywords: adelic general linear group
zbMATH Keywords: \(L\)-function
zbMATH Keywords: irreducible admissible generic representation
zbMATH Keywords: entire functions
zbMATH Keywords: automorphic representation
author: James Cogdell
author: I. I. Pyatetskii-Shapiro
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: mrl.1996.v3.n1.a7
OpenAlex ID: W2045018110
DOI: 10.4310/MRL.1996.V3.N1.A7
description: scientific article; zbMATH DE number 913096