On the positivity of the central critical values of automorphic \(L\)-functions for \(GL(2)\)
type: Item
instance of: scholarly article
title: On the positivity of the central critical values of automorphic \(L\)-functions for \(GL(2)\)
zbMATH Open document ID: 0861.11032
DOI: 10.1215/S0012-7094-96-08307-6
author: Jiandong Guo
published in: Duke Mathematical Journal
publication date: 1996-11-10T00:00:00Z
review text: Let \(F\) be a number field, \(A\) be the adeles of \(F\), and let \(\pi\) be a cuspidal automorphic representation of \(GL(2,A)\) with trivial central character. Let \(\eta= \prod \eta_v\) be a quadratic idele class character of \(F\) such that \(\eta_v=1\) at all archimedean places \(v\). Then the author shows that the \(L\)-value \(L(1/2,\pi) L(1/2, \pi \otimes \eta)\) is nonnegative. Here \(1/2\) is the center of the critical strip. As the author notes, combining this theorem with results on the positivity of an average of \(L(1/2,\pi \otimes \eta)\) as \(\eta\) varies, which should be obtained by combining standard analytic methods with the results of the reviewer and \textit{J. Hoffstein} [Ann. Math., II. Ser. 142, 385-423 (1995; Zbl 0847.11026)], this theorem implies that the central value \(L(1/2,\pi)\) itself is nonnegative. Nonnegativity is consistent with the Riemann Hypothesis for the \(L\)-function \(L(s,\pi)\) -- indeed, if \(L(1/2, \pi)\) were negative, then there would be a real zero of \(L(s, \pi)\) strictly between \(1/2\) and 1. The main ingredient in the proof of this theorem is an identity which expresses the value \(L(1/2, \pi) L(1/2, \pi \otimes \eta)\), up to a product of local factors at bad places, as the square of the norm of certain period integrals. These integrals are periods, over non-split tori coming from the quadratic extension \(E\) of \(F\) determined by \(\eta\), of the cusp forms in certain Jacquet-Langlands correspondents of \(\pi\). This kind of formula was first obtained by \textit{J.-L. Waldspurger} [Compos. Math. 54, 173-242 (1985; Zbl 0567.10021)], using the theory of the Weil representation. However, the author notes that Waldspurger's result is not sufficient for this application. Instead, the author establishes his result by further developing the relative trace formula introduced by \textit{H. Jacquet} [Ann. Sci. Éc. Norm. Supér, IV. Ser. 19, 185-229 (1986; Zbl 0605.10015)] to study Waldspurger's result. On one side of…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 912124
zbMATH Keywords: positivity of the central critical values
zbMATH Keywords: automorphic \(L\)-functions for \(GL(2)\)
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: \(L\)-value
zbMATH Keywords: cuspidal automorphic representation
zbMATH Keywords: relative trace formula
zbMATH Keywords: matching orbital integrals
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 results of Atkin and Lehner
cites work: Formes Modulaires et Representations De GL(2)
cites work: Linear periods.
cites work: Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: Q4052229
cites work: Sur un résultat de Waldspurger
cites work: On the nonvanishing of some \(L\)-functions
cites work: Automorphic forms on GL (2)
cites work: On Euler Products and the Classification of Automorphic Representations I
cites work: Heegner points, cycles and Maass forms
cites work: Fourier coefficients and modular forms of half-integral weight
cites work: Values of L-series of modular forms at the center of the critical strip
cites work: Q4132638
cites work: Nonvanishing of \(L\)-functions for \(\mathrm{GL}(2)\)
cites work: The special values of the zeta functions associated with Hilbert modular forms
cites work: On the critical values of certain Dirichlet series and the periods of automorphic forms
cites work: On the fundamental periods of automorphic forms of arithmetic type
cites work: On the Fourier coefficients of Hilbert modular forms of half-integral weight
cites work: Q3900137
cites work: Correspondances de Shimura et quaternions
cites work: Q3682555
cites work: Sur les coefficients de Fourier des formes modulaires de poids demi-entier
full work available at URL: s0012-7094-96-08307-6
OpenAlex ID: W1537397571
description: scientific article; zbMATH DE number 912124