On the non-vanishing of the central value of certain L-functions: Unitary groups

Dihua Jiang; Lei Zhang

doi:10.4171/JEMS/955

On the non-vanishing of the central value of certain L-functions: Unitary groups

Dihua Jiang, Lei Zhang

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Let π be an irreducible cuspidal automorphic representation of a quasisplit unitary group U_n defined over a number field F. Under the assumption that π has a generic global Arthur parameter, we establish the non-vanishing of the central value of L-functions, L(1/2, π × χ), with a certain automorphic character χ of U₁, for the case of n = 2, 3, 4, and for the general n ≥ 5 by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central L-values by means of the theory of endoscopy.

Original language	English (US)
Pages (from-to)	1759-1783
Number of pages	25
Journal	Journal of the European Mathematical Society
Volume	22
Issue number	6
DOIs	https://doi.org/10.4171/JEMS/955
State	Published - 2020

Bibliographical note

Funding Information:
The research of the first named author is supported in part by the NSF DMS-1600685 and DMS-1901802, and that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore.

Publisher Copyright:
© European Mathematical Society 2020

Keywords

Automorphic L-functions
Automorphic descents
Bessel periods
Central values
Endoscopic classification of discrete spectrum
Fourier coefficients of automorphic forms
Unitary groups

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10.4171/JEMS/955

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title = "On the non-vanishing of the central value of certain L-functions: Unitary groups",

abstract = "Let π be an irreducible cuspidal automorphic representation of a quasisplit unitary group Un defined over a number field F. Under the assumption that π has a generic global Arthur parameter, we establish the non-vanishing of the central value of L-functions, L(1/2, π × χ), with a certain automorphic character χ of U1, for the case of n = 2, 3, 4, and for the general n ≥ 5 by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central L-values by means of the theory of endoscopy.",

keywords = "Automorphic L-functions, Automorphic descents, Bessel periods, Central values, Endoscopic classification of discrete spectrum, Fourier coefficients of automorphic forms, Unitary groups",

author = "Dihua Jiang and Lei Zhang",

note = "Funding Information: The research of the first named author is supported in part by the NSF DMS-1600685 and DMS-1901802, and that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore. Publisher Copyright: {\textcopyright} European Mathematical Society 2020",

year = "2020",

doi = "10.4171/JEMS/955",

language = "English (US)",

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pages = "1759--1783",

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T1 - On the non-vanishing of the central value of certain L-functions

T2 - Unitary groups

AU - Jiang, Dihua

AU - Zhang, Lei

N1 - Funding Information: The research of the first named author is supported in part by the NSF DMS-1600685 and DMS-1901802, and that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore. Publisher Copyright: © European Mathematical Society 2020

PY - 2020

Y1 - 2020

N2 - Let π be an irreducible cuspidal automorphic representation of a quasisplit unitary group Un defined over a number field F. Under the assumption that π has a generic global Arthur parameter, we establish the non-vanishing of the central value of L-functions, L(1/2, π × χ), with a certain automorphic character χ of U1, for the case of n = 2, 3, 4, and for the general n ≥ 5 by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central L-values by means of the theory of endoscopy.

AB - Let π be an irreducible cuspidal automorphic representation of a quasisplit unitary group Un defined over a number field F. Under the assumption that π has a generic global Arthur parameter, we establish the non-vanishing of the central value of L-functions, L(1/2, π × χ), with a certain automorphic character χ of U1, for the case of n = 2, 3, 4, and for the general n ≥ 5 by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central L-values by means of the theory of endoscopy.

KW - Automorphic L-functions

KW - Automorphic descents

KW - Bessel periods

KW - Central values

KW - Endoscopic classification of discrete spectrum

KW - Fourier coefficients of automorphic forms

KW - Unitary groups

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JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 6

ER -

On the non-vanishing of the central value of certain L-functions: Unitary groups

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