Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives
type: Item
instance of: scholarly article
title: Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives
zbMATH Open document ID: 0699.10039
DOI: 10.2307/1971508
author: Daniel Bump
author: Solomon Friedberg
author: Jeff Hoffstein
published in: Annals of Mathematics. Second Series
publication date: 1990-01-01T00:00:00Z
review text: Let f be a cusp form on hyperbolic 3-space, transforming according to one of the two characters of \(PGL_ 2({\mathbb{Z}}[i])\) that are trivial on the principal congruence subgroup modulo \((1+i)\). Suppose that f is a Hecke eigenform. For the twisted automorphic L-functions \(L(s,f,\chi)\) it is proved: 1) If Re \(s\geq 1/2\) there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) such that \(L(s,f,\chi)\neq 0.\) 2) If the character of \(PGL_ 2({\mathbb{Z}}[i])\) is non-trivial there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) for which \(L(1/2,f,\chi)=0\) and \(\frac{\partial L}{\partial s}(1/2,f,\chi)\neq 0.\) The proof is complicated, and contains many computations. The reviewer admires the insight of the authors in deciding what automorphic forms to use and what to compute. The basis idea is to consider Eisenstein series on \(G=GSp_ 4({\mathbb{C}})\) for certain congruence subgroups \(\Gamma\), and to compute Fourier coefficients. Actually, these Eisenstein series live on the metaplectic group. That means here that they are functions on G transforming according to a certain character of \(\Gamma\), the Kubota symbol. The existence of this symbol and the necessary properties are proved. The Eisenstein series considered here are connected to a maximal parabolic subgroup, with \(GL_ 2\) as the Levi component. So they depend on one complex parameter s, and on an automorphic form on \(GL_ 2({\mathbb{C}})\), as which the cusp form f is chosen. Whittaker-Fourier coefficients are now computed, not only for the Eisenstein series themselves, but for various left translates as well. A suitable parametrization of the flag variety for G is developed to describe explicitly the Dirichlet series that are the main factor of these coefficients. It is shown that in the region of convergence certain linear combinations of these Dirichlet series happen to be of the form \(L(s-1,f,\chi)\) up to some explicitly known factors. One of…
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 4146011
zbMATH Keywords: nonvanishing of L-functions
zbMATH Keywords: twisted L-functions
zbMATH Keywords: Grassmannians
zbMATH Keywords: Whittaker functions
zbMATH Keywords: twisted automorphic L-functions
zbMATH Keywords: metaplectic group
zbMATH Keywords: Kubota symbol
zbMATH Keywords: Eisenstein series
zbMATH Keywords: flag variety
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971508
OpenAlex ID: W2330992598
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description: scientific article; zbMATH DE number 4146011