On the Ramanujan conjecture and finiteness of poles for certain \(L\)-functions
type: Item
instance of: scholarly article
title: On the Ramanujan conjecture and finiteness of poles for certain \(L\)-functions
zbMATH Open document ID: 0654.10029
DOI: 10.2307/2007005
author: Freydoon Shahidi
published in: Annals of Mathematics. Second Series
publication date: 1988-01-01T00:00:00Z
review text: Let \(G\) be a quasi-split reductive algebraic group over a number field. Let \(P\) be a maximal parabolic subgroup of \(G\) and \(M\) a Levi component of \(P\). Given a cuspidal representation \(\pi\) of \(M\), one constructs an Eisenstein series on \(G\), which depends on one complex parameter s. There is a well-known expression for the constant term of this Eisenstein series, in which occur the (partial) \(L\)-functions attached to \(\pi\) and some finite-dimensional representations \(r_ i\) of the \(L\)-group of \(M\). The author gives a list of the highest weights of the representations \(r_ i\) extending Langland's original list. Under the assumption that \(\pi\) is generic the author proves that the products defining these \(L\)-functions are absolutely convergent for \(\text{Re}(s)>2\). Under more restrictive assumptions local \(L\)-functions are defined for the ramified places and it is proved that the thus completed \(L\)-function extends to a meromorphic function on \({\mathbb C}\) with only finitely many poles and satisfies a functional equation. There is an induction argument in the proofs. This induction is possible thanks to the important Lemma 4.2, whose proof involves case by case checking. The method of proof also gives an estimate for the eigenvalues of Hecke operators on generic cusp forms on certain quasi-split groups.
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 4067015
zbMATH Keywords: automorphic L-functions
zbMATH Keywords: Fourier coefficients of Maass wave forms
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: quasi-split reductive algebraic group
zbMATH Keywords: cuspidal representation
zbMATH Keywords: Eisenstein series
zbMATH Keywords: constant term
zbMATH Keywords: finite-dimensional representations
zbMATH Keywords: highest weights
zbMATH Keywords: local L-functions
zbMATH Keywords: finitely many poles
zbMATH Keywords: functional equation
zbMATH Keywords: eigenvalues of Hecke operators
zbMATH Keywords: generic cusp forms
zbMATH Keywords: quasi-split groups
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2007005
OpenAlex ID: W1991502429
Wikidata QID: Q122957356
description: scientific article; zbMATH DE number 4067015