Symmetric square \(L\)-functions on \(GL(r)\)
type: Item
instance of: scholarly article
title: Symmetric square \(L\)-functions on \(GL(r)\)
zbMATH Open document ID: 0753.11021
DOI: 10.2307/2946548
author: Daniel Bump
author: David Ginzburg
published in: Annals of Mathematics. Second Series
publication date: 1993-01-16T00:00:00Z
review text: This paper is devoted to a study of the \(L\)-functions associated with automorphic representations of \(GL(r,k_ \mathbb{A})\) and the `symmetric square' representation of the \(L\)-group \(GL(r,\mathbb{C})\) of \(GL(r)\). The analytic continuation and functional equation of this function have already been established through the theory of Eisenstein series by Shahidi but this method has some serious inherent restrictions, especially in regard to the analysis of the poles and zeros of the \(L\)- function. Here the authors give an ingenious generalization of a construction of Patterson and Piatetski-Shapiro which gives a Rankin- Selberg representation of these \(L\)-functions. It involves an integral of Shimura's type. The interesting feature here is that neither the `theta function' on \(GL(r)\), or rather its double cover, nor, for \(r>2\), the one on \(GL(r-1)\) used to construct the Eisenstein series has a Whittaker model in the usual sense. However both have certain unique degenerate Whittaker models and it is this that the authors exploit. This method leads to a new proof of the meromorphic continuation and of the functional equation. As is to be expected the method does not quite yield poles and zeros of the \(L\)-function, only when a certain number of factors are left out. However with this proviso the authors obtain very sharp results about the poles, which are rather exceptional.
reviewed by: Samuel James Patterson
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 90885
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: automorphic representations
zbMATH Keywords: symmetric square
zbMATH Keywords: \(L\)-group
zbMATH Keywords: Rankin-Selberg representation
zbMATH Keywords: integral of Shimura type
zbMATH Keywords: degenerate Whittaker models
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: functional equation
zbMATH Keywords: poles
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946548
OpenAlex ID: W2324493103
description: scientific article; zbMATH DE number 90885