Table Of ContentZETA FUNCTIONS OF
REDUCTIVE GROUPS
AND THEIR ZEROS
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b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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ZETA FUNCTIONS OF
REDUCTIVE GROUPS
AND THEIR ZEROS
Lin Weng
Kyushu University, Japan
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Library of Congress Cataloging-in-Publication Data
Names: Weng, Lin, 1964– author.
Title: Zeta functions of reductive groups and their zeros / by Lin Weng
(Kyushu University, Japan).
Description: New Jersey : World Scientific, 2018. | Includes bibliographical
references and index.
Identifiers: LCCN 2017053916 | ISBN 9789813231528 (hardcover : alk. paper)
Subjects: LCSH: Functions, Zeta. | Linear algebraic groups.
Classification: LCC QA351 .W46 2018 | DDC 515/.56--dc23
LC record available at https://lccn.loc.gov/2017053916
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Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
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Dedicated to my wife and
our daughter and son
v
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Introduction
AsanaivegeneralizationoftheDedekindzetafunctionofanumberfield F,
forapositiveintegern,thereisanaturalintegration
(cid:90)
Θ(Λ)vol(Λ)sdµ(Λ) (cid:60)(s)>1. (0.1)
Mtot
F,n
Here, Mtot denotes the moduli space of (O -)lattices Λ of rank n (with O the
F,n F F
integerringof F),Θ(Λ)andvol(Λ)denotethethetaseriesandtheco-volumeof
Λ,respectively.Ifitwereconvergentwhenn≥2,(0.1)wouldbeviewedasarank
nnon-abelianzetafunctionofF,sincethisintegrationrecoverstheDedekindzeta
functionofF ifn=1,andtheranknlatticesareassociatedtothegroupGL ,for
n
whichonlyGL iscommutative. Obviously,usingtheMellintransform,(0.1)can
1
berewrittenas
(cid:90) ∞(cid:90) dT
Θ(Λ)vol(Λ)sdµ(Λ)
T
0 Mtot[T]
F,n
(cid:90) (cid:32)(cid:90) ∞ dT(cid:33)
= Θ(T21n ·Λ)Ts dµ(Λ) (0.2)
T
Mtot[1] 0
F,n
(cid:90)
= E(cid:98)(Λ,s)dµ(Λ),
Mtot[1]
F,n
where, for T > 0, Mtot[T] denotes the moduli space of rank n lattices of co-
F,n
volumesT > 0, and E(cid:98)(Λ,s)denotesthe(complete)EpsteinzetafunctionsofΛ.
SinceEpsteinzetafunctionsarespecialkindsofEisensteinseriesandhenceare
well-knowntobeofslowgrowth,andMtot[T]arenotcompact,e.g. inthecase
F,n
F = Q,Mtot[1]isisomorphictoSL (Z)\SL (R)/SO ,alltheintegrationsabove
F,n n n n
(overmodulispaces)diverge.
Thefirsttaskofthisbookistoremedytheaboveconstructionstoobtaincon-
vergentintegrations. MotivatedbyMumford’sstabilityinalgebraicgeometry, it
isonlynaturaltotruncatethemodulispacesMtot[T]usingthestabilitycondition
F,n
so as to obtain the compact moduli spaces M [T] and hence also the moduli
F,n
spaceM ofsemi-stablelatticesofrankn.
F,n
vii
December21,2017 12:32 ws-book9x6 BC:10723-ZetaFunctionsofReductiveGroupsandTheirZeros WengZeta pageviii
viii Introduction
Definition0.1. Theranknnon-abelianzetafunction(cid:98)ζ (s)ofanumberfield F
F,n
isdefinedby
(cid:90)
(cid:98)ζ (s):= Θ(Λ)vol(Λ)sdµ(Λ) (cid:60)(s)>1. (0.3)
F,n
M
F,n
Theorem 0.1. Non-abelian zeta functions satisfy all basic zeta properties.
Namely,
(0) UptoaconstantfactordependingonnandthelocalandglobalunitsofF,
(cid:98)ζ (s)=(cid:98)ζ (s).
F,1 F
(1) Theintegration(0.3)definesahomomorphicfunctioninsandadmitsaunique
meromorphiccontinuation(cid:98)ζ (s)tothewholecomplexs-plane.
F,n
(2) Zetafunction(cid:98)ζ (s)satisfiesthestandardfunctionalequation
F,n
(cid:98)ζ (1−s)=(cid:98)ζ (s).
F,n F,n
(3) Zeta function(cid:98)ζ (s) has two singularities only, i.e. two simple poles at s =
F,n
0, 1,andtheresiduesadmitthefollowinggeometricinterpretations
Ress=1(cid:98)ζF,n(s)=vol(cid:0)MF,n[1](cid:1).
One of the central themes of this book is to expose algebraic, analytic and
geometric structures of these zeta functions. In particular, a weak Riemann hy-
pothesisfortheranknnon-abelianzetafunctionwillbeestablished,ensuringthat
allbutfinitelymanyzerosof(cid:98)ζQ,n(s)lieonthecentralline(cid:60)(s)=1/2whenn≥2.
Toexplainthis,asin(0.2)above,theMellintransformtransforms(0.3)into
(cid:90)
(cid:98)ζF,n(s)= E(cid:98)(Λ,s)dµ(Λ). (0.4)
MF,n[1]
Since the Epstein zeta function E(Λ,s) coincides with the Eisenstein series
ESLn/Pn−1,1(1,g;s), induced from the constant function one on the Levi subgroup
of the maximal subgroup P of SL corresponding to the ordered partition
n−1,1 n
n = (n−1)+1,theranknnon-abelianzetafunctionmaybeviewedasaspecial
geometric Eisenstein period, defined as the integration of Eisestein series over
somecompactmodulispaces.
Naturallyassociatedtozetafunctions,Eisensteinseriesplayacentralrolein
numbertheoryaswell.Fromtheirtheories,rangingfromtheclassicalSiegeltothe
modernLanglands,andthetraceformulatechniques,oftheprimitiveSelbergand
thepowerfulArthur,foranL2automorphicfunctionϕonaLevisubgroupMofa
parabolicsubgroupPofareductivegroupG,withrespecttoasufficientlyregular
December21,2017 12:32 ws-book9x6 BC:10723-ZetaFunctionsofReductiveGroupsandTheirZeros WengZeta pageix
Introduction ix
parameterT,thereisaninducedEisensteinseriesEG/P(ϕ,g;λ)onG(F)\G(A)1/K
andhenceananalyticEisensteinperiod
(cid:90)
(cid:16) (cid:17)
ΛT EG/P(ϕ,g;λ) dg. (0.5)
G(F)\G(A)1/K
HereAdenotestheadelicringof F, Kdenotesamaximalcompactsubgroupof
G(A), and ΛT denotes the Arthur’s analytic truncation. In general, it is difficult
tocalculateananalyticEisensteinperiod. However, whenϕiscuspidal, thereis
a way to evaluate, thanks to the advanced Rankin-Selberg formula obtained by
Jacquet-Lapid-Rogowski with a use of regularized integrations over cones. For
example,whenP= Bisminimalandϕistheconstantfunction1onthemaximal
torus, based on the structures of the constant terms of EG/B(1,g;λ) exposed by
Siegel in lower rank and Langlands in general, it is proved that (0.5), for suffi-
cientlyregularT,coincideswiththeperiodinthefollowing:
Definition0.2. TheT-periodofasplitreductivegroupG overanumberfield F
isdefinedby
(cid:88) e(cid:104)wλ−ρ,T(cid:105) (cid:89) (cid:98)ζ ((cid:104)λ,α(cid:105))
ωG;T(λ):= (cid:81) F . (0.6)
F w∈W α∈∆(cid:104)wλ−ρ,α(cid:105)α>0,wα<0(cid:98)ζF((cid:104)λ,α(cid:105)+1)
HereW,∆,ρ,>and<0denotetheWeylgroup,thesetofsimpleroots,theWeyl
vectorandthepositiveandnegativerootsintheinducedrootsystem,respectively.
Inaddition,byLanglands’theoryonEisensteinsystems,beinginducedfrom
a special L2-automorphic form, the Eisenstein series ESLn/Pn−1,1(1,g;s) admits in
principlearealizationasmultipleresidueofsomeEisensteinseriesinducedfrom
cusp forms over Levi subgroups of some higher co-rank parabolic subgroups.
Practically,inthisbook,asanSL -analogueofaresultofDiehl,thefollowingex-
n
plicitrealizationofthesinglevariableESLn/Pn−1,1(1,g;s)isobtainedasamultiple
residueofaseveralvariablesESLn/P1,...,1(1,g;λ),whereP1,....1denotestheminimal
parabolicsubgroupofSL associatedtothepartitionn=1+···+1.
n
Lemma 0.1. For SL , denote by (cid:8)λ (cid:9)n−1 the fundamental dominant weights.
n i i=1
(cid:88)n−1
Then,withλ= sλ +ρ∈Cn−1ands= s ,
i i n−1
i=1
(cid:16) (cid:17)
ESLn/Pn−1,1(1,g;s)=Ressn−2=1,...,s2=1,s1=1 ESLn/P1,...,1(1,g;λ) . (0.7)
As a by-product, up to a certain normalization factor Norm(s) to effectively
collectthezetafactorsinthedenominatorsofthetermsontherighthandsideof
(0.8)below,thereisthefollowing:
