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ZETA FUNCTIONS OF REDUCTIVE GROUPS AND THEIR ZEROS 10723_9789813231528_TP.indd 1 23/1/18 9:21 AM b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM 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 111111111111111111111111111111000000000000000000000000000000777777777777777777777777777777222222222222222222222222222222333333333333333333333333333333______________________________999999999999999999999999999999777777777777777777777777777777888888888888888888888888888888999999999999999999999999999999888888888888888888888888888888111111111111111111111111111111333333333333333333333333333333222222222222222222222222222222333333333333333333333333333333111111111111111111111111111111555555555555555555555555555555222222222222222222222222222222888888888888888888888888888888______________________________TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP..............................iiiiiiiiiiiiiiiiiiiiiiiiiiiiiinnnnnnnnnnnnnnnnnnnnnnnnnnnnnndddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd 222222222222222222222222222222 2222222222222233333333333333//////////////11111111111111//////////////1111111111111188888888888888 99999999999999::::::::::::::2222222222222211111111111111 AAAAAAAAAAAAAAMMMMMMMMMMMMMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit http://www.worldscientific.com/worldscibooks/10.1142/10723#t=suppl Printed in Singapore LaiFun - 10723 - Zeta functions of reductive groups.indd 1 19-12-17 9:39:38 AM November15,2017 15:15 ws-book9x6 BC:10723-ZetaFunctionsofReductiveGroupsandTheirZeros WengZeta pagev Dedicated to my wife and our daughter and son v b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM January9,2018 12:12 ws-book9x6 BC:10723-ZetaFunctionsofReductiveGroupsandTheirZeros WengZeta pagevii 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: