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cmcsc10ElmarGrosse-Kli*onne{L|{Y cmr8ReceivÎed: AugustX6,2002 yRevised: JanÎuaryX11,2003OjOCommÎunicatedXbyPeterSchneider'QE Abstract.01LetlR3bGetheringofintegersinaniteextensionKqof
msbm10Q 0er cmmi7pR,letkbGeitsresidueeldandlet:ٓR cmr71|s(X )
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cmsy10!Rǟ^O! cmsy7
`Ͳ=GL1(RDz)bGe a>"geometric"rankonerepresentationofthearithmeticfundamental group6ofasmoGothanekP-schemeX .jW*eshowthattheloGcallyK - analytice>characters:Rǟ^ !C^፴p %?aree>theCpR-vqaluedpGointsofaK -rigid spaceUUW'Ҳandthat
Hs2L(8;Tc)=՟ u
cmex10Y7W W xO2X<$K*1w fe oA V18 ()(FrGobwV W x)Trdeg
tur(W W x) y;= viewedasatwovqariablefunctioninT*and,#eismeromorphicon A^1lqy msbm7C O
\ cmmi5psjW }.ZOnMthewayMweprove,basedonaconstructionofW*an,
C aslopGedecompositionforordinaryoverconvergent(niterank)[ٲ- moGdules,UUintheGrothendieckgroupofnuclear[ٲ-moGdules. 2000WMathematicsSub 8jectClassication:vPrimary14F30;XSecondary 14G10,UU14G13,14G15,14G22 IntroductionInjaseriesofremarkqablepapGers[html:14 html:
][html:15 html:][html:16 html:],#W*anrecentlyprovedalongoutstanding2 conjectureofDworkonthep-adicmeromorphiccontinuationofunitroGotL-functionsarisingfromanordinaryfamilyofalgebraicvqarietiesdened]overaniteeldkP.W*ebGeginbyillustratinghisresultbyaconcreteexample.Fixn0andletYI̲bGetheanen+1-dimensionalFpR-vqarietyinA^1S8G^n+1፴mֲdenedUUbyqzppHɸ 8z7=x0S+:::g+:::/xnq~:DeneuՍ:Yq!Gm7bysending(zp;x0|s;:::;xnq~)tox0|sx1'|jxnq~.XF*orrՍ1andy"2F䍷p rletܵYy·=Fp rbGethebreofuabovey[ٲ.QF*orm1letYy·(Fp r,rm)bGetheset DocumentuUa#Mathematica8(2003)1{42 *e̍6XIhtml: html:2t(ElmarGrosse-Kli*onne+4e̍6XIofCFp r,rm-rationalpGointsand(Yy·)0?thesetofclosedpointsofYy·=Fp rjL(aclosed 6XIpGointǵz9^isanorbitofanC fe rF9p rB-vqaluedpointunderthep^rm-thpowerF*robeniusmap6XIp r ;4its$~degreedegH
qƴr(zp)isthesmallestpGositiveintegerdsuchthat^[ٴdp rˇxesthe6XIorbitUUpGointwise).qThezetafunctionofYy·=Fp r^isP!W>Z (Yy·=Fp r ;Tc)=exp7(ݷ1ckXm=1<$yjYy(Fp r,rm)jyw fe ,s (֍mB#Tcm*)=
YjzI{2(Y yU) Z cmr50<$91"Sw fe 3h c.18 Tcdeg
turr|,(zI{)Vt:Ex6XIOnUUtheotherhandforacharacter :Fpfj!CUUdenetheKloGostermansumLlmKm(y[ٲ)=kX͍r
ex1i*0n cmsy52! msbm5Fqƍnpr,rmᎍx 0 x 1 x nl=y0T/ (T*r
N9:F6³pr,rm^k=F p*(x0S+8x1+:::g+xnq~))(^6XIandUUletL (Y 9;Tc)bGetheseriessuchthat GTcdlogL (y[;Tc)=f1*Xm=1^Km(y)Tcm*:m6XIThen,UUasseries,
qˍ wY7 c bL (Y 9;Tc)=Z (Yy·=Fp r ;Tc);6~6XIhenceOtounderstandZ (Yy·=Fp r ;Tc)weneedtounderstandalltheL (y[;Tc).6XISuppGosem isnon-trivial.ItisknownthatL (y[;Tc)isapolynomialofdegree6XIn8+1:qthereUUarealgebraicintegers0|s(y[ٲ);:::;nq~(y)UUsuchthatvoL (y[;Tc)( 1)rn 1!|ɲ=(18 0|s(y)Tc)(1 nq~(y)Tc): 6XIThese\iTL(y[ٲ)have\complexabsolutevqaluep^r7n=25andare`-adicunitsforany6XIprime}`b6=p.W*easkfortheirp-adicvqaluationandtheirvariationwithy[ٲ.6XIEmbGeddingC fe QҸ!C fe fpQp!$wehaveiTL(y[ٲ)2QpR()where^p 1= p.CSpGerberhas6XIshownthatwemayordertheiTL(y[ٲ)suchthatordٟp+(iTL(y[ٲ))=iforany0in.6XIFixUUsuchaniandfork2ZconsidertheL-functionڍ Yj zy@L2(G m) 0 =F p<$ 1 6w fe K f18 : zk;Zi
(y[ٲ)Tcdeg
tur1[(y@L) #6XI(heredegΞqƱ1K(y[ٲ)istheminimalr,suchthatyѸ2UF䍷p r ,}and(Gm)0|s=FpJaisthesetof6XIclosed
}pGointsofGm=Fpϲdenedsimilarlyasbefore).XA
jpriorithisseriesdenes6XIa;holomorphicfunctiononlyontheopGenunitdisk.%Dworkconjecturedand6XIW*anbprovedthatitactuallyextendstoameromorphicfunctiononA^1lC p ,and6XIvqaries/uniformlywithkinsomesense.ePNowletWlbGetherigidspaceoflocally6XIQpR([ٲ)-analyticcharactersofthegroupofunitsintheringofintegersofQpR([ٲ).6XIInUUthispapGerweshowthat {rL(T V;)=`Yjy@L2(G m) 0 =F p<$[,12&w fe W c.18 (iTL(y[ٲ))Tcdeg
tur1[(y@L) DocumentuUa#Mathematica8(2003)1{42
e̍6XIhtml: html:GOnFUamiliesofPureSlopeL-Functions>ϲ3+4e̍6XIdenesdameromorphicfunctiononA^1lC p
W }.=SpGecializing2Wtodthecharac-Ae6XItercr%7! rG^kfork.2ZwerecoverW*an'sresult.Theconceptualwaytothinkof 6XIthisUexampleisintermsof[ٲ-moGdules:sFpKactsonYݲviaz87!)z+9Mafora2FpR.6XIIt
inducesanactionofFpontherelativen-thrigidcohomologyR^nq~ur7ig@L;ٸOY of6XIu,FandBoverQpR([ٲ)thelattersplitsupintoitseigencompGonentsforthevqarious6XIcharactersofFpR.The -eigencompGonent(R^nq~ur7ig@L;ٸOYG)^
(6isanoverconvergent
V6XI[ٲ-moGduleandL (y;Tc)^( 1)rn 1#\sisthecharacteristicpGolynomialofF*robenius6XIactingonitsbreiny[ٲ.*CrucialistheslopGedecompositionof(R^nq~ur7ig@L;ٸOYG)^ :6XIit meansthatforxeditheiTL(y[ٲ)vqaryrigidanalyticallywithy{insomesense.6XIW*eLLarethusledtoconsiderDwork'sconjecture, i.e.VW*an'stheorem,inthe6XIfollowingUUgeneralcontext.6XILetJ2R]bGetheringofintegersinaniteextensionKNofQpR,iletbeauni-6XIformizerkkandktheresidueeld.LetX4MbGeasmoothanekP-scheme,pletAbe6XItheecoGordinateringofaliftingofX.qtoasmoothaneweakformalRDz-scheme6XI(soAisawcfg-algebra)andletx䍑bAbGethep-adiccompletionofA../LetcbeanRDz-6XIalgebra:DendomorphismofAliftingtheq[ٲ-thpGower:DF*robeniusendomorphismof6XIX ,xwhereq"=jkPj.VVAniterank[ٲ-moGduleoverx䍑_vbA
(resp.overA)isaniterank
@ 6XIfreex䍑bnrA
s-moGdulenr(resp.A-module)togetherwitha[ٲ-linearendomorphism.A6XInite"rank[ٲ-moGduleoverx䍑ub"A
Eis"calledoverconvergent"ifitarisesbybasechange 6XIA!x䍑#bA
e/fromaniterank[ٲ-moGduleoverA._]Lettheniterankoverconvergent6XI[ٲ-moGdule#Roverx䍑ǫb#RA
ƥbe#Rordinary*,-SinthestrongsensethatitadmitsaFrobGenius6XIstable,ltrationsuchthatonthej -thgradedpiecewehave:/ttheF*robGeniusis6XIdivisible/by[ٟ^jandmultipliedwith[ٟ^ j
嵲itdenesaunitroGot[ٲ-modulej6, ji.e.6XIa[ٲ-moGdulewhoselinearizationisbijective.(Recallthatunitroot[ٲ-modules6XIoverx䍑_^5εA
띲are5thesameascontinuous5representationsof1|s(X )onniterankfree6XIRDz-moGdules.){AlthoughXlisoverconvergent,Y2jwillXlingeneralnotbeovercon-6XIvergent;irand
hthisiswhatprevented
hDworkfromprovingwhatisnowW*an's6XItheorem:5theL-functionL(j6;Tc)ismeromorphiconA^1lC p .IMoreoverheproved6XIthe
sameforpGowers
(=iteratesofthe[ٲ-linearendomorphism)^k;ZjEofjDaand
t<6XIshowedthatincasejŔisofrankonethefamilyfL(^k;Zj됵;Tc)gk+B2Zvqariesuniformly6XIwith!kX2RZinacertainsense.m,AttheheartofW*an'sstrikingmethoGdlieshis6XI"limiting 4[ٲ-moGdule"constructionwhichallowshimtoreducetheanalysisof6XIthe/notnecessarilyoverconvergent/j۲tothatofoverconvergent/[ٲ-moGdules|6XIatthecostofnowworkingwithoverconvergent[ٲ-moGdulesofinniterank,
but6XIwhichparenuclear.hT*othelatterageneralizationoftheMonskytraceformula6XIcan8IbGeappliedwhichexpressesL(^k;Zj됵;Tc)asanalternatingsumofF*redholm6XIdeterminantsUUofcompletelycontinuousDworkopGerators.6XIThe;5rstaimofthispapGeristofurtherexplorethesignicanceofthelimiting6XI[ٲ-moGdule9sconstructionwhichwethinktobGerelevqantforthesearchofgoGod6XIp-adicDcoGecientsonvqarietiesincharacteristicp.@F*ollowinganargumentof6XIColeman[html:4 html: ]wegiveafunctorialityresultforthisconstruction.dThisisthen6XIusedtoprove(Theoremhtml:7.2 html:)aslopGedecompositionforordinaryoverconvergent
@ 6XIniterank[ٲ-moGdules,
intheGrothendieckgroup(x䍑\ubA)ofnuclear[ٲ-moGdules6XIoverx䍑7bA
[.I Moreprecisely*,weshowthatanyjƲasabGove,notnecessarilyovercon- DocumentuUa#Mathematica8(2003)1{42 *e̍6XIhtml: html:4t(ElmarGrosse-Kli*onne+4e̍6XIvergent,canbGewritten,in(x䍑\ubA),asasumofvirtualnuclear ':
cmti10over}'convergent 6XI[ٲ-moGdules.\(Thisistheglobalversionofthedecompositionofthecorrespond-6XIingɵL-functionfoundbyW*an.)DOursecondaimistostrengthenWan'suniform6XIresultstonthefamilyfL(^k;Zj됵;Tc)gk+B2Znincasejjisofrankone.Moregenerally6XIwe>replaceju1bytherankoneunitroGot[ٲ-moduledet"(j6)ifju1hasrank>1.
p6XILetdetljlbGegivenbytheactionof2x䍑bA^onabasiselement. F*or~ fe gx
A2X6XIaclosedpGointofdegreef2:letx:x䍑sbAM!Rf M˲beitsT*eichmullerlift,Q whereRf 6XIdenotesUUtheunramiedextensionofRiofdegreef.qThen6B 8wwV W x=x( z[ٲ():::f 1m())6XIliesinRǟ^Ȳ.{W*eprovethatforanyloGcallyK -analyticcharacterZ:Rǟ^
.Ÿ!C^፴p6XItheUUtwistedL-functionk卒 IL( z;T V;)=՟Y7W W xO2X<$81w fe H֟ V18 (wV W x)Tcrdeg
tur(W W x)6XIisõp-adicmeromorphiconA^1lC p ,Uandvqariesrigidanalyticallywith. >More
C6XIprecisely*,kBbuilding3onworkofSchneiderandT*eitelbaum[html:13 html:
],kBweuseLubin-6XIT*ate{theorytoconstructasmoGothCpR-rigidanalyticvqarietyWNSwhoseCp-vqalued6XIpGointsQareinnaturalbijectionwiththesetHom&K}-״an-y(Rǟ^ȵ;C^፴p)oflocallyK -6XIanalyticUUcharactersofRǟ^Ȳ.qThenourmaintheoremis:6XIhtml: html::Theorem[0.1.OntheCpR-rigidsp}'aceA^1lC pѸWo:thereexistsameromorphic
̍functionsL
%whosepul lb}'acktoA^1lC p4viaA^1lC pW![NA^1lC pڏW };UPt7!(t;)sforany
J2HomqK}-ian,-(Rǟ^ȵ;C^፴p)=W }(CpR)isac}'ontinuationofL( z;T V;).ThestatementintheabstractabGovefollowsbythewellknowncorrespGondencebGetweenkrepresentationsofthefundamentalgroupandunit-roGot[ٲ-modules.TheX$analyticvqariationoftheL-seriesL( z;T V;)X$withtheweightX$makesitmeaningfulR.tovqastlygeneralizetheeigencurvethemestudiedbyColemanandMazurؑ[html:5 html: ]inconnectionwiththeGouv^Gea-Mazurconjecture.zNamely*,`wecanasks