Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
type: Item
instance of: scholarly article
title: Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
zbMATH Open document ID: 0847.11026
DOI: 10.2307/2118638
author: Solomon Friedberg
author: Jeff Hoffstein
published in: Annals of Mathematics. Second Series
publication date: 1995-11-22T00:00:00Z
review text: Let \(\pi\) be a cuspidal automorphic representation of \(\text{GL}(2, \mathbb{A})\) and let \(L(s, \chi\otimes \pi)\) denote the standard \(L\)-function of \(\pi\) twisted by a character \(\chi\). The following is proved. If \(\pi\) is not equivalent to its dual, there are infinitely many quadratic characters \(\chi\) such that \(L({1\over 2}, \chi\otimes \pi)\neq 0\). Here, the local components of \(\chi\) may be prescribed at a finite number of places. If \(\pi\) is equivalent to its dual, either the same is true or there are infinitely many \(\chi\) such that \(L(s, \chi\otimes \pi)\) has a simple zero at \(s= {1\over 2}\). The proof uses a Dirichlet series with twisted \(L\)-functions as coefficients and which is obtained from a Rankin-Selberg integral involving Eisenstein series on the double cover of \(\text{GL}(2, \mathbb{A})\). A brief history of nonvanishing results for these \(L\)-functions is given in the introduction of the paper.
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 819353
zbMATH Keywords: nonvanishing theorems
zbMATH Keywords: cuspidal automorphic representation
zbMATH Keywords: quadratic characters
zbMATH Keywords: twisted \(L\)-functions
zbMATH Keywords: Rankin-Selberg integral
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118638
OpenAlex ID: W2011047812
description: scientific article; zbMATH DE number 819353