Periods of automorphic forms: the case of \((\mathrm{U}_{n+1}\times \mathrm{U}_{n},\mathrm{U}_{n})\)
type: Item
instance of: scholarly article
title: Periods of automorphic forms: the case of \((\mathrm{U}_{n+1}\times \mathrm{U}_{n},\mathrm{U}_{n})\)
zbMATH Open document ID: 1430.11066
author: Atsushi Ichino
author: Shunsuke Yamana
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2019-01-09T00:00:00Z
review text: In this paper the authors compute regularised period integrals of automorphic forms associated to unitary groups. More precisely, they compute the regularised periods of certain cuspidal Eisenstein series as well as their residues. As an application, the authors make further progress on the Gan-Gross-Prasad conjecture for unitary groups. Recall that this conjecture gives a representation theoretic criterion for non-vanishing of the central critical value of a certain Rankin-Selberg automorphic \(L\)-function [\textit{W. T. Gan} et al., Astérisque 346, 1--109 (2012; Zbl 1280.22019)]. The original Gan-Gross-Prasad conjecture for unitary groups as well as a more refined version has been proved by \textit{W. Zhang} [Ann. Math. (2) 180, No. 3, 971--1049 (2014; Zbl 1322.11048); J. Am. Math. Soc. 27, No. 2, 541--612 (2014; Zbl 1294.11069)] using the relative trace formula of \textit{H. Jacquet} and \textit{S. Rallis} [Clay Math. Proc. 13, 205--264 (2011; Zbl 1222.22018)], under certain local conditions on the intervening representations. Using their period computations, the authors are able to relax these local conditions (see Theorem 5.1 for the precise statement).
reviewed by: Nils Matthes
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 6999121
zbMATH Keywords: periods
zbMATH Keywords: Gan-Gross-Prasad conjecture
MaRDI profile type: Publication
full work available at URL: crelle-2015-0107
OpenAlex ID: W2469205718
cites work: Multiplicity one theorems
cites work: Q3964660
cites work: The trace formula in invariant form
cites work: On the inner product of truncated Eisenstein series
cites work: Polar Decomposition for P-adic Symmetric Spaces
cites work: Canonical Extensions of Harish-Chandra Modules to Representations of G
cites work: An analogue of the Cartan decomposition for \(p\)-adic symmetric spaces of split \(p\)-adic reductive groups
cites work: Harmonic analysis in weighted L2-spaces
cites work: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
cites work: Nonvanishing of the central critical value of the third symmetric power L-functions
cites work: On the nonvanishing of the central value of the Rankin-Selberg 𝐿-functions
cites work: Q5482673
cites work: Q5325595
cites work: Q3009492
cites work: The refined Gross-Prasad conjecture for unitary groups
cites work: On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture
cites work: Periods of automorphic forms: the case of
cites work: Q4662303
cites work: Q3183923
cites work: Periods of automorphic forms
cites work: Automorphic forms on \(GL(3)\). I, II
cites work: Q3019009
cites work: On Euler Products and the Classification of Automorphic Representations I
cites work: Uniqueness of Bessel models: the archimedean case
cites work: A product of tensor product \(L\)-functions of quasi-split classical groups of Hermitian type
cites work: Square integrability of representations on \(p\)-adic symmetric spaces
cites work: Q3019010
cites work: Q4681585
cites work: Q5493436
cites work: Periods of Eisenstein series: the Galois case
cites work: Q4263342
cites work: Q4854091
cites work: Endoscopic Classification of representations of Quasi-Split Unitary Groups
cites work: Arithmetic of Hermitian forms
cites work: Multiplicity one theorems: the Archimedean case
cites work: Q4004200
cites work: PERIODS OF AUTOMORPHIC FORMS: THE TRILINEAR CASE
cites work: Periods of residual automorphic forms
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
cites work: Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups
cites work: Automorphic period and the central value of Rankin-Selberg L-function
DOI: 10.1515/CRELLE-2015-0107
description: scientific article; zbMATH DE number 6999121