Sums of twisted GL(2) \(L\)-functions over function fields.
type: Item
instance of: scholarly article
title: Sums of twisted GL(2) \(L\)-functions over function fields.
zbMATH Open document ID: 1048.11039
DOI: 10.1215/S0012-7094-03-11735-4
author: Benji Fisher
author: Solomon Friedberg
published in: Duke Mathematical Journal
publication date: 2003-11-17T00:00:00Z
review text: This paper concerns a weighted sum of twisted \(L\)-functions, \[ \sum_D L(s, \pi \otimes \chi_D) a_0(s, \pi, D) \eta(D) | D| ^{-w}. \] Here \(\pi\) (respectively, \(\eta\)) is a cuspidal automorphic representation of \(\text{GL}_2(\mathbb{A}_K)\) (respectively, \(\text{GL}_1(\mathbb{A}_K)\)) and \(\chi_D\) are quadratic characters. The field \(K\) is a function field of a smooth projective curve \(C\) over a finite field of odd characteristic. The sum runs over a certain set of effective divisors of \(C\). The weight functions \(a_0(s, \pi, D)\) have been defined by \textit{D. Bump}, \textit{S. Friedberg} and \textit{J. Hoffstein} [Sums of twisted GL(3) automorphic \(L\)-functions, in Contributions to automorphic forms, geometry and arithmetic, Johns Hopkins Univ. Press, Baltimore, MD, 131--162 (2004)], who proved the uniqueness. That is, there is a unique family of weight functions such that the sum satisfies a certain nonabelian group of functional equations. This paper presents a different approach. The authors rewrite the sum as a multiple Dirichlet series and give a new description of the weight functions. They prove the sum has a finite, nonabelian group of functional equations. They also establish a similar result in the noncuspidal case, where the natural object is a sum in three complex variables. The main result is the proof that the sums are rational functions. The authors specify the possible denominators and the degrees of the numerators of these rational functions.
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11M38
zbMATH DE Number: 2005279
zbMATH Keywords: twisted \(L\)-function
zbMATH Keywords: double Dirichlet series
zbMATH Keywords: automorphic forms
MaRDI profile type: Publication
cites work: Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives
cites work: Nonvanishing theorems for L-functions of modular forms and their derivatives
cites work: On some applications of automorphic forms to number theory
cites work: Local Rankin-Selberg convolutions for 𝐺𝐿_{𝑛}: Explicit conductor formula
cites work: Q5532660
cites work: Multiple Dirichlet Series and Moments of Zeta and L-Functions
cites work: Mean values of \(L\)-functions over function fields
cites work: Double Dirichlet series over function fields
cites work: Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
cites work: Average values of L-series in function fields.
cites work: Q3995750
cites work: Introduction to Lie Algebras and Representation Theory
description: scientific article; zbMATH DE number 2005279