; TeX output 2012.10.07:0942x=]XQ cmr12DiagonalhrypSersurfacesandtheBlocrh-KatoconjectureaSirPreterSwinnerton-Dyer j'81.@ cmti12IntrffoductionfǹTheresearcrhrepSortedinthispapSerre ectsadesireto'study&theL-functionassoSciatedwith'thenonsingularhrypersurfaceg cmmi12W=URW2cmmi8m;n'in!N cmbx12P2m lde nedorverQwhicrhisgivenbytheequationyQa|{Ycmr80X nڍ0 D{+:::+a̾mĽX nڍm Z=UR0|(1)'whereImUR>1andn>2.5Herewrecantakethea̾itobSenonzerointegers.5The'onlysucrhL-functionswhichareinterestingarethoseinmiddledimension.'The~bSeharviourof}theseatintegralvXaluesof}sisthesubjectoftheTVatecon-'jectureT|andT{theBloScrh-Katoconjecture.ThoughbSothofthesewrereoriginally'statedabSoutvXarieties,itappearsthattheyarebothreallyaboutmotivres.8LetVO`bSeacompletenonsingularvXarietryde nedoverQ.Theoutward'andvisiblesignofamotivreassoSciatedwithVfaisasubspaceofthecohomol-'ogyKspaceJwhicrhismappSedintoJitselfbyeveryJFVrobSeniusmap.yHencethere'is`anL-seriesattacrhed_toanymotive,;andanyL-seriesbSelonging_toV<йisthe'proSductoftheL-seriesbelongingtotheirreduciblemotivresassociatedwith'Vʴin.Ethe.DappropriatecoSdimension.WVecallamotivregeffometricUifthecorre-'spSondingcohomologyspaceH s̻geom%ԹisspannedbrytheclassesofsubrvXarieties'of!Vde ned orver:Q<,dandtrffanscendentalWif thecorrespSondingcohomology'subspacehastrivialinrtersectionwithH ̻geom l.FVoranysuchVp,)theTVatecon-'jecture primarilyconcernsthe geometricmotivres;theirL-seriesarebSelieved'toBbSeBmrultiplicativecombinationsofBthezeta-functionsofalgebraicnrumbSer' elds.The~BloScrh-Kato~conjecturedescribes~bothsortsof~motivre,{butshould'probablyJbSeKsplitupaccordingasthemotivreisgeometricortranscendenrtal.'Thetranscendenrtalpartisthemorenovel.8Inthisparagraph,-letL(s)bSeanrymotivicL-functionanddtheassociated'coSdimension./It$is#knorwnthattheseriesforL(s)isabsolutelyconrvergent$in'!", cmsy10Fu1z@2 d+1;anditisconjecturedthatL(s)canbSemeromorphicallyconrtin-'uedtothewholes-planeandthatthereisafunctionalequationconnecting'L(s)andL(dl+m1s).AccordingtotheBloScrh-Katoconjecture,.ifsUR>Fu1z@2 (d+l1)'isinZthenthereisafairlywrellde nedtranscendenrtalTusuchthatٽL(s)=T'isinQ;moreorver@:L(s)UR=TFG!'where?fG?gistheorderofacertainsetalecohomologygroupandF-(thefudge'factorJ)VisaproSductofTVamagarwaVnumbers.ThereVisa nitesetofinrteger1*x='vXalueswofs,calledwstrffategicvalues,forwwhicrhtheconjecturedformulaforT'islesscomplicatedthanitisingeneral.D(FVorageometricmotivredhasto'bSeevren,&andtheonlystrategicvXalueiss=Fu616z@2 dV͹+V1.)./fA?moSdi edvrersion'of0the0BloScrh-Katoconjecture,{forwhendisoSddands=Fuڻ1ڟz@2 '(dʹ+1),{takres'someCzaccounrtCyofthefactthatL(s)maryvXanishthere;{4thisappSearstobSemore'orMlesscompatiblewiththeBircrh/Swinnerton-DyerMconjecture.aButMforthe'latterconjecture,AMwhenL(Fu33133z@2j(dA+1))isnon-zeroFistheproSductofTVamagarwa'nrumbSers/witha.certainnon-localfactor.߸It.isingeneraldiculttoobtainanry'arithmetic information abSouttheorderofacohomologygroup.VTheTVate-'Safarevicgroupisanexception,GbutthisisbSecausewrealsoharvealow-brow'description2ofit.ItwrouldbSe2good2tohavesimilar2low-browdescriptionsof'thegroupswhicrhoSccurintheBlocrh-Katoconjecture.8The%motivicL-functionsassoSciatedwithhrypersurfaces%oftheform(1)are'HecrkeCL-functions.ETheobjectofthisCpapSeristocomputeandinrterpretL(s)'atIstrategicJvXaluesofsfortranscendenrtalmotivesJofsuchJhypSersurfaces;þthe'samemethoSdscanbeusedtocomputeL(s)atnon-strategicvXaluesofs,but'IhLamhmnothlabletoinrterpretthese..IhLhavehlcomputedL(s)onlyforthevXalues'nq=r4;5RandR10,lbutIRharveRnoreasontodoubtthatthesearetrypical.qThe'nrumericalevidenceimpliesthatL(s)UR=TRwhereӽTBisatranscendenrtalwhich'canbSeexactlyspeci edandRisarationalnrumbSerwhosedenominatorisa'pSorwerof2;whennUR=4thiscanprobablybSeprorvedbyargumentslikethose'inR[2]Sbutconsiderablymorecomplicated.%WhennUR=5Ror10,1therationalitry'ofwR0canvpresumablyagainbSeprorvedbywmeansofthetheoryofcomplex'mrultiplicationPasdevelopSedbyShimuraandhispupils;:$butanystatement'abSoutthedenominatorofRwroulddependonconsiderableadvXancesinthat'theoryV.8The͚casewhen͛disoSddandsUR=Fu1z@2 (ddl+1)͚isofparticularinrterest,bSecauseit'isTthenSplausibletohopSethattheGintheBloScrh-KatoconjectureshouldbSea'square.*?Indeed,ǍinalltheexampleswhicrhIhavecomputedRistheproSduct'ofasquareandapSorwerof2.](Asimilarphenomenon,K\thoughassociated'withaquitedi erenrtvXarietyV,wasalreadyrepSortedin[4].)Inthemaintable'conrtainedinthispapSer,ktherearetwoexampleswhereL(s)hasazeroof'pSositivreMevenorder.ThewholebSehaviourissosimilarLtowhathappSensin'theparticularcasedescribSedbrytheBirch/Swinnerton-Dyerconjecturethat'oneisbSoundtowronderwhetherinthismoregeneralsituationthereisan'AbSelianвvXarietrywithгaroletoplayV.MoreoverthereisгanobviousAbSelian'vXarietryinthewings.2-x='Question1lDoffes35theintermediateJacobianplayaroleinthetheory?J'Again,Kthereisoneobrviousreasonwhytheorderofa niteabSeliangroup'mighrtbSecompelledtobeasquare:thisisifitadmitsanon-singularskrew-'symmetricbilinearform.+(Itiswrell-knownthatthishappSensfortheTVate-'Safarevicgroup,thebilinearformbSeingtheCasselsform.'Question2lWhen5,sX=Fu)1)z@2 `(d+1),5istherffeanaturalskew-symmetricbilinear'form35onthecffohomology35groupappearingintheBloch-Katoconjecture?8I[gam[deeply[indebtedtoJohnTVateforagreatdealofhelpfuladvice,'withoutwhicrhthispapSercouldnothavebSeenwritten.982.%Prffeliminaries!FVorCanryDnUR>2,let̾n=e22I{i=n逹andwriteDko=Q(̾nP)with'ringofinrtegers$%n eufm10o.rLetv/=22 if2jjn,andv/=v0otherwise.LetpbSeaprime'of@Z?whicrhdoSesnotdividenoranryofthea̾idڹ,sothatWRhasgoSod@reductionx*l~'W76anon-zeroelemenrtofZ=nisde nedasʍgn9(rS)UR=XǍx2"2@cmbx8Fq% cmsy6G;cmmi6qQ rb(x) (x):'Ѝ'AnryYotherXnontrivialadditiveXcharacter n920˹hasXtheform n9201wede netheJacobisumtobSelv5Jr(r;p)UR=J(r̻0;:::ʜ;r̾mĹ;p)UR=qn9 1 ʵgn9(r̻0)g(r̾mĹ)l'wherether̾iOsatisfyBr̻0j+:::+r̾m ZUR0 moSd&6n; n6jr̾idڽ:'InS=particularSqH?A(m)+1%{7'wherethebracrketisabinomialcoSecient.XByinductiononm,A(m)isa'pSolynomialV>inV=nofdegreeatmostm.gItisthereforeenoughto ndaformrula'for꨽A(m)undertheadditionalhrypSothesisthatnisprime.8FVoranrycinZand xedprimenletN@(c)bSethenumbSerofsetswithl읽r̻0j+:::+r̾m ZURc moSd&6n; 0L-functionofWlastheproSductofsimplerL-functions.'Apartfrom̾k#(s(m1)=2)whenmisoSdd,eacrhofthemhastheformໍyM@(s)UR=Y ҍ%\%eufm8p 0x(1Jr(r;p)eU\n(a rϹ)(Norm p) s ) 1 \|:&M'(WVeusetheletterM6sinsteadofLtoindicatethatwrehaveleftoutpSossible'factors correspSondingtothe badprimes.) FVorsucrhanL-functiontobSe'geffometricyit3his3gnecessarythatmisoSddand (cr)UR=Fu1z@2 (m4\1)for3hallc,Xsothat'eacrhz>characteristicroSotatz=pisqn92(m1)=2&7timesarootz=ofunitry; conjecturally'thisisalsosucienrt. Onecasewhenthisconditionholdsiswhen,after'pSermrutation(ofther̾iifnecessaryV,&r̻0+r̻1f/=,r̻2+r̻3=,:::=+n.`FVor(surfaces'withٽnT<1000thisistheonlycase,ʥandIbSelievrethistobSesoingeneral.'This4is3closelylinkredtotheassertion(whicrhhasbSeenprorved4insomecases;'see]forexampleShioSda[4])thatinmiddledimensionthegeometricpartof'thecohomologyisgeneratedbrytheobviouslinearsubspaces.983.%Evaluation$ofcffertainJacobisumsɹFVoranryidealbinoprimetonand'the^a̾i89wrecan_nowde ne_Jr(r;b)by_multiplicativityV.1Weil^shows_thatJr(r;b)'consideredIasaJfunctionofbisaGrossencrharacterandn22lMisade ningideal'for/it.* FVor xedrandnaformrulaforJr(r;b)canthereforebSeobtainedbya' niteGcalculation;~#andGsucrhformulaeGforsmallvXaluesofnaregivreninBerndt'etal[1].9TVatehasobtainedageneralformrulaforJr(r;b),Wbutforpresenrt'purpSosesthefollorwingsimplerresultswillsuce.'Lemma2cuQSuppffosebthatn}=P(orbn=2P'wherebP(isanoddprime, and'that35qË=URp2 fǺwitheven.fiThenJr(r;p)=(1)2m+1@qn92(m1)=2&.and(a2rϹ)=1.6Ix='PrffoofESincey~AiszthesmallestpSositivrevXalueofsforwhicrhp2s (u1moSd":n,'wre9musthave8qn921=2b=p2=21moSd":n.%WVehave n9(x)=2TJr ,z2(x)for9some'nonrtrivialNp2th rroSotofMunityn9,5wherethetraceisforF̾q=F̾pϪandF̾pϫisidenti ed'withZ=(p).8SinceGal(F̾q=F̾p])isgeneratedbryxUR7!x2p,foranryr_gn9(rS)UR=XUT(x prٿ) (x p])=XUT(x prٿ) (x)=g(rSp):m'Inparticulargn9(rS)=g(rSp2=2 )=g(rS),dandwreknowthatgn9(rS)g(r)='2rb(1)qn9.8If꨽nUR=PnisanoSddprime,thengn9(rS)URX'؍x6=0UT (x)1 moSd&6(1);(D'soZwremusthave2rb(1)UR=1YandZgn9(rS)=q21=2 E.IfZn=2P2 then,jmoSdY(154),% f1gn9(rS)URX'؍x6=0UV((x)) P.:r $. (x)z(ǭ g(PrS):nif꨽r>6isoSddn;fa 1:nif꨽r>6isevrenq:'IfQrLisoSddand n9(x)UR=2TJr ,z2(x)cthenQg(PrS)isaquadraticGausssum,pSwhicrhis'equalto(s2)23p2=2whereȄ=UR1ifpUR1moSd":4andȄ=URiifpUR3moSd":4.'Thrusgn9(rS)iscongruentto1moSd(1)ifrD4isoSddandp__3moSd":4and'2jjǹ,and8to19otherwise.Sogn9(rS)8isq21=2~inthe9formercaseandqn921=2~inthe'latter. Since *theformer +casehappSensanevrennumbSeroftimesintheproduct'for꨽Jr,thisprorvestheformrulaforJ.8Sincep1isprimetoPƹ,evreryelementofF̾pisaPƟ2th1wpSorwer;andsince'oisevren,everyelementofF̾pisasquareinF̾q.8So(a2rϹ)UR=1.<'Lemma3cuQIf35nUR=PisanoffddprimethenJr(r)1moSd":(1)22.'PrffoofFVortianrytjxinF2RAq 4de neh(x)inZ=(n)bry(x)UR=2h(x)Q.vFVorthistjLemma'=S=UR꨹andso|-X((x)) rUR1rSh(x)(1) mod&6(1) 2;'thereforemHgn9(rS)UR1+r(1)XUTh(x) n9(x) mod&6(1) 2:@'Thrus꨽Jr(r)UR1moSd":(1)22.}7Y,x='Lemma4cuQSuppffose35thatr̻0;r̻19areintegersnotdivisiblebyn.fiThen$U_jÀgn9(r̻0)g(r̻1)UR=z(ǭ ((1))2rq0qyPifnj(r̻0j+r̻1);fa gn9(r̻0j+r̻1)Jr2v(r̻0;r̻1)yPotherwiseu;%'wherffe35Jr2v(r̻0;r̻1)UR=P(x2rq0(1x)2rq1).'PrffoofӹWhen@Jnj(r̻0+Nr̻1)theresulthasalreadybSeen@Iprorved;yso@Jwecanassume'that꨽nUR6j(r̻0j+r̻1).8NorwDn-0m8Í(qgn9(r̻0)g(r̻1)UR=XX(((xyn9)) rq0((x)) rq1 (x(1+y))G*+q=URX((1)) rq0((x)) rq0*+rq1N+XPSX'؍yI{6=1-q6UqōJxy [z ΍1+y'oq0ǟqw<>:| 1if35r̻0;r̻19arffebotheven,fa ((2))22rq1if35r̻09isoffddandr̻1even, (1)((2))22(rq0*+rq1)if35r̻0;r̻19arffebothodd..t_'Prffoofm̹AsDinCtheproSofofLemma3,Wwrite(x)UR=2h(x)Q.7SuppSoseC rstthatr̻1'isevren.8Let̻0V=UR0ifr̻0isevenbut̻0V=URPnifr̻0isoSdd.8Ifx6=0wrehave[((1x)) rq1 URf1Fuۻ1۟z@2 Qr̻1(1)h(1x)g moSd&6(1) 28 fx='and((1x))2rq1 UR1moSd":(1).8Since((x))2rq0 UR((x))2q0moSd+4(1),FJr v(r̻0;r̻1)UR1+Fuۻ1۟z@2 Qr̻1(1)v@X'؍x6=0;1Aٹ((x)) q0 h(1x) moSd&6(1) 2:'k卑'Inparticular,ifr̻0iisalsoevrenthenJr2v(r̻0;r̻1)UR1moSd":(1&&)22.#PFVor xedy'inF̾qthenrumbSerofsolutionsofx(1x)UR=yXis1+2P̹(14yn9);so*B3=3$GͽJr v(rr;rS)p{O=URXUV((x(1x))) r紹=XUVf1+ P̹(14yn9)g((y)) r3=p{O=URXUV((4yn9)) rb P̹(14y)=(4 rb)UR=Jr v(rr;Pƹ)=(4 r):'5ލ'TVaking꨽r=UR2givresJr2v(2;Pƹ)(224)moSd":(1)22andthereforeSp˟Xd#((x)) P̽h(1x)UR(1(2 4))=(1) moSd&6(1):'Henceifr̻1isevrenandr̻0isoSddthent Jr v(r̻0;r̻1)UR1Fuۻ1۟z@2 Qr̻1(1(2 4)) moSd&6(1) 2'andthereforeN*Jr v(r̻0;r̻1)UR((2)) 2rq1moSd3E(1) 2:8If꨽r̻0;r̻1arebSothoddthen.0d[EAJr v(r̻0;cX(r̻1)UR=v@X'؍x6=0;1Aٹ((x 1 \|) rq0((1x 1)) rq1!⍍fz=URXUV((x)) rq0*rq1((x1)) rq1 =(1)Jr v(r̻0jr̻1;r̻1)3:+0c'Usingthepreviousresult,thiscompletestheproSofoftheLemma.)'CorollaryhIfAno=2PcthenJr(r)((1))2s=2 D((2))22R3moSd-(1ST)22 wherffe's˺isthenumbfferofoffddr̾j gֺandRisthesumoftheevenr̾jf .,Ifkhhasclass'numbffer351wecanchooseUR1moSd":(1)229and35then?Jr(r)UR=((1)) s=2 D((2)) 2R 3:'PrffoofWhenNnmUR=1Nothisisthe rstcaseinLemma4.Thegeneralresultthen'follorwsbyinductiononm,usingLemmas4and5.xk9 vx='Lemma6cuQSuppffose'thatn=4'andp=q1moSd":4.CVLet'p=(n9)with'ËUR1moSd":2(1+i)35bffeaprimefactorofpinko=Q(i).fiThen#k &dlpJr v(1;1)UR=(1)n9;Jr (3;3)=(1)E;lpJr v(1;2)UR=Jr (2;1)=n9;Jr (2;3)=Jr (3;2)= m: R'Prffoofs1WVeharvel]s0Jr v(1;2)UR=XUV(1(x))(1 2(1x))+p:l^'InkthislsumthetermswithxuA=uB0;1vXanishandinevreryothertermthe rst'factorIjisdivisibleIibry1aP+aOiandthesecondbyIi2.!SoJr2v(1;2)UR1moSd":2(1aO+aPi).'Since(Jr2v(1;2))UR=pasideals,Jr2v(1;2)=n9.8FVorK xedKӽy inF̾p1thenrumbSerKofsolutionsofx(1f>x)UR=y is1f>+22(1f=4yn9);'so&0 ÍU+Jr v(1;1)~=URXUT(x(1x))=XUVf1+ 2(14yn9)g(y)3=~=URXUT(4yn9) 2(14y)=(4)UR=Jr v(1;2)=(4)'and꨽(4)UR=1.8TheremainingclaimsfollorwbycomplexconjugacyV.'CorollaryhSuppffose35thatrcontainsn̾j?copiesofjforj%=UR1;2;3,35sothatkyFu1z@2 (mر+1).'WhenmUR=4or6,˱wrecananalyticallycontinueM@(s)toFu1z@2 (m[[1)ina'wray which,CforcertainintegervXaluesofs,Cexpressesit asa nitesuminterms 10 x='offWVeierstrassfellipticfunctions;form'J=4ffulldetailscanbSefoundbSelorw.'Butq theq correspSondingprocessq forothervXaluesofminrtroSducesfunctions'relatedtoAbSelianvXarietiesofdimensionFu616z@2 (mNwithcomplexmrultiplicationby'o,cLwhereNXܹistheorderofGal(kg=Q)V1(Z=(n))2;andthetheoryofsucrh'functionsisnotyretexplicitenoughtomeetourneeds.'WVethereforeharveto'proSceedinadi erenrtwayV.8WVebSeginbryobtainingthefunctionalequation, usingthemethoSdsof'HecrkeWbratherthanWathoseofTVate'sthesis. Onereasonforthiscrhoiceisthat'Hecrke'smethoSdgivresmoreinformationaboutthethetaseries.8De ne bryJr(r;( O))qd ~a2r V(W* Lm)EKq$ n-E=UR ( )Y ҍncUV(c ) (rc-:1 )?;"@'so/that. ( O)asafunctionofa2r isaDiricrhletcharacter.whosevXaluesare'pSorwers of andn22QWa̾i9isade ningidealforit.Theonlypropertryof 'that(mņŅ1)wrethenassignctoC̾mjv1c¹.?ThisleavesinlimbSothecfor'whicrhBcisBinC߻(m1)=2!;ntheyshouldbSethoughtBofasnearlybutnotquitein'C߻(m1)=2!.8Inthisnotationwrehave2&`4L(s)z4h=URX% |b ( O)(Norm M ) s Y`  f( n91ڍc ʵ ) jf ( n91ڍc ) m1jcg z4h=URX% |b ( O)YUX \f( n91ڍc ʵ ) m12j!ƹ( n91ڍc : n91ڍc ) jvsg:.)'FVorƈsimplicitrywenowassumethatk-hasclassnumbSer1.,(IfthishypSothesis'isedroppSed,theargumenrtwhichefollowsonlyneedssomeefurthernotational'complications;butHMIGseenoHLpSoinrtinextendingthecomputationsto elds'withclassnrumbSergreaterthan1.)8WVriteZE$# 0(y0;v;f)UR=X ҍh0 2oexp(*f2n7XCmr+Íy̾c.yd̾c(̾c(fG )+v̾c.y)(c (f )+vc )g#'wherethey̾c andd̾c arerealandpSositivre. WVeshallcrhoSosethed̾c later.'Theav̾c|willsatisfyav̾cM?=şPp2ctRAn(Du̾t͹wheretheu̾t̹arereal;0thrusweacanidentify'theUspaceUofvwiththespaceofu,pIwhicrhisR2nP.yInparticularweUshalluse'thisidenrti cationtodetermineameasureonthev0-space.Since#20 isnot'a ectede2brye3multiplyingf1bye3aunit,itdoSesnotdepSendonthecrhoiceoffG.'Thegseriesfor#20, likrealltheanalogousthetaserieswhichweshallconsider,'isd}absolutelyd|conrvergentandd|uniformlysoprorvidedy|andvlieind|bSounded'domains[&andeacrhy̾cisbSoundedawayfrom0.YThisisenoughtojustifyall'themanipulationswhicrhfollow.8LetJbSethelatticeofpoinrtsuJinQ2N cforwhichtheassoSciatedvzaresuch'that8|eacrhv̾cfis8{inf."\Thusv̾cD=ʽ̾c.yv̻1forsuchvh|and#20is8{pSeriodic8|inuwith'asitslatticeofpSeriods.8LetdUR=(s2)bSethedi erenrtofkg;thusJ#dUR=(nYĚpjnUV(1exp*(2n9i=p)) 1 \|): 12 x='FVor('ink Ewith̾c=x̾c.y,P̾cv̾cԡisin(Zforeacrhv(assoSciatedwithsome'pSoinrtofifandonlyifisin(df)21 \|.8Thus#20hasaFVourierexpansion># 0V= vXUR2(df)1'b()expf2n9iX3CUT̾c.yv̾cg:('If꨽VisthevrolumeofR2ND=then3tlj8b()UR=Vp 1甆Z# 0expf2n9iXUT̾c.yv̾cgdun=URVp 1甆Zexp/lf2n9(X%Cmr+UVy̾c.yd̾cv̾cvc u+iX3CUT̾cv̾c)gdu1'wherethe rstinrtegralistakenoverR2ND=andthesecondintegralistaken'orver!YR2ND.Inthelastinrtegralwereplace!Zeachv̾cOҹbyv̾c]ic =yjcjadjcj.Doing'somshiftseacrhintervXalm(1;1)ofintegrationbyanimaginaryconstant,but'bryastandardargumentwecanshiftitback;henceweobtain`b()UR=Vp 1 Iexpf2n7XCmr+Í̾c.yc =y̾cd̾cg%>'wherezz]IFչ=UR甆ZMUT1 1|:::+g甆ZM7g121CDexpUf2n7XCmr+Íy̾c.yd̾cv̾cvc gdu:'This'2inrtegralcan'3bSeevXaluatedexplicitly;Ewbutallwreneedtoknow'3isthatit'isequaltoaconstanrtpSositivemultipleofQ?(y̾c.yd̾c)21 \|.8Thus]&Ž# 0V=URVp 1 IX ҍ F׹exp(:Yf2n7XCmr+Í̾c.yc =y̾cd̾c!+2n9iX3CUT̾cv̾cg:|(5)%㒍'This'whererȽW̻1( )UR=X ҍ xv (W* Lmvn9)expf2iTVr (W* v=fGs2)g'withmvHrunningthroughasetofrepresenrtativesmoftheclassesin(o=f)2.Since' Jisprimetof,,thereisonesucrhW* khv4ineacrhclassof(o=f)2;9%sowrecanwrite'thelastequationas5W̻1( )UR=X ҍ xv (vn9)expf2iTVr (v=fGs2)g%'wherevZMrunsthroughasetofrepresenrtativesof(o=f)2.='HereW̻1( )doSes'depSendonthecrhoiceofIandfG.ExplicitlyV,ifwremultiplyIorfbryaunit'wre$*alsomultiply$)W̻1.by ().fThe$)function#alsodepSendsonthecrhoiceof's2,butinamorecomplicatedwrayV.8OnthelefthandsidewreobtainJiX'؍P} ] ( O)expf2n7XCmr+Íy̾c.yd̾c(̾c )(c )gYUX \(yjcjadjcjc ) m2jv1 14,x='wherethesumisorverallelemenrts =URfG 7+vXofo.8Thusifwewritee܍?S(y0; )UR=X'؍ h  ( O)expf2n7XCmr+Íy̾c.yd̾c(̾c O)(c )gYUX \(c ) m2jv1%'I'thenwrehave6S(y0; )YUX \yOn9m2j yjcjչ=URW(y 1 |; )|(7)$'forUmsomeW3indepSendenrtofthey̾c.y,pwherey021isthevectorofthey2n91RAc "forc'inC52+ϭ.8Inotherwrords,replacingybyy021w$multiplies FS(y0; )YUX \yߍn9m=2j 뀍jcj'bry꨽Wƹ.8ItfollowsfromthisthatWƟ22 =UR1.8Infactwehave#>`4W=URVp 1 Iz qYCmr+ٽy̾c.yz!3/ :ZYI Np(i̾c.y(fGs2)djcja) 2jv+1m!Ɵ  GW̻1( ):$'G8TheMellintransformformrulagives6,甆ZMB.1=ػ0M4exp_f2n9yd̾c.y(̾c O)(c )gyn9 sjō*dy*[z C ΍ y!^=UR(sjӹ)f2n9d̾c.y(̾c O)(c )g jvs n'for꨽cinC52+ϭ.8ThenprorvidedFu1z@2 (m+1),!rySpL(s)YUX ō4A(sjӹ)H[z4* ΍(2n9djcja)sjM>=UR甆ZMUT1 0UZ*UV甆ZM6UX110AU^S (y0; )YUX Zq경yOn9sj yjcjt*dyjcj*_z ΍ yjcj&fq|(8) W"'whereh:S (y0; )UR=X% |b ( O)expf2n7XCmr+Íy̾c.yd̾c(̾c O)(c )gYUX \(c ) m2jv1!ƽ:$'WVe~norw}moSdify(8)insuch}awayas}toreplaceS2by} |thatis,toreplacethe'sumMorverMallbbryasumorverMall O.Letthes2(c) forcinC52+ VbSemrultiplicatively'indepSendenrt+totally+realunitsinkg,=URgn9 1 ʳ甆ZʳS(y0; )YUX Zq경yOn9sj yjcjt*dyjcj*_z ΍ yjcj&fq!''where>the=inrtegralistaken=over(0;1)aV̻1.gNow=split>therangeofinrtegration'forz)ҹastheunionofthetrwosubintervXals(0;1)and(1;1)anduse(7)on'theformer;thenwreobtain2Ѝ[rgWHL(s)_Z Ync ōyL(sjӹ)t[z4* ΍(2n9djcja)sj:I=!_Z gn9 1 ʳ甆Zʳq$ʴS(y0; )YUX \yOn9sj yjcjt*dyjcj*_z ΍ yjcj)-+W(y; )YUX \yOn9msj yjcjtfdyjcjf_z ΍ yjcj4q#;|(9)0|'wheretheinrtegralisnowtakenover(1;1)V̻1.Therighthandsideprovides'theanalyticconrtinuationofؽL(s)tothewholes-plane,dandwrealsoobtain'thefunctionalequation.8Thelattersarysthat 6eL(s)YUX ō4A(sjӹ)H[z4* ΍(2n9djcja)sj!1{'ismrultipliedbyWnifwereplacesbyms.8Horwever, themultipleintegralwhichwehaveobtainedforýL(s)in(9)is'notusuallyconrvenientfornumericalcalculation, 4anditisbSettertoproceed'in;aO =O2 20G3qP 4X=URqd * 20 ǽ q[ 4&(1) -:0Ǿ=4; qdN1+ih .Iq7š 4?=URi (1-:2*)=4; ; qdk2N "q&n 4/ Ĺ=i =2:!8TVo|-simplify|.theinrterpretationofthetranscendenrtalfactorofL(s)for'strategicp˽s,UitisadvXanrtageousptomakepuseofWVeierstrassfunctions. JLet'!=2o6220575543:::'bSe\sucrhthattheperiods\oftheWVeierstrassfunction'satisfying=}20w^2r=4}2354}=are!n9;i!.The=mostconrvenient=waytoevXaluate'}(z)isbrymeansoftheformula#KōKb!n922.=}(z)Kb[z# ΍(2n9i)2s'=ōQ[z+ ΍(1Q)24"+P1X ҍr ΍ (1qn9rQ1 \|)3`Oq;;/KōPN!n9(z)PN[zƟ ΍2n9irb=Fu1z@2 dōZ1۟[z ΍1Q%ōMiz۟[z @ ΍2!Z+P1X ҍr1isapSositivreintegerwhich'isfourthpSorwerfree,itisshorwnin[2]thatŽa 1=4 @ L aϹ(1)=! .1and*U(4a) 1=4L a G(1)=!'areu)rationalinrtegers,andu(theirvXaluesareextensivelytabulatedthere.bThe'relevrentpartoftheBircrh/Swinnerton-Dyerconjectureisasfollorws:I\L aϹ(1)UR=(!n9a 1=4܄)FX 2'whereX X22 isX!theorderoftheTVate-SafarevicgroupandFl=FƟ2aȟQF̾p }where'the proSductistakrenover theoSddprimesdividinga.gFVorsuchp F̾p 2=?2,'whicrh>istheTVamagawanumbSerofthecurveyn922=UR4x23Wrm4ax, andFƟ2 CisFu4s1qz16if'aisasquareandFuP1Pz@4Fotherwise.0(ThrusFqisalmostbutnotquiteaproSductof'loScalfactors.)8RecallalsothattheTVate-SafarevicgroupisanHV21Z.8The0case1m=3isthe1oneconsideredin[3];Jwresummarizeherethe'relevrent]partoftheconclusions. Thetrwo]tetradswhicrhgiverisetotranscen-'denrtalL-functionsare(1;1;1;1)and(3;3;3;3),aanditisenoughtoconsider'theC rstBofthese.Denotebryathefourth-pSorwer-freepartCofa̻0a̻1a̻2a̻3 GandS'write4(a)=qd -a a.q 4 i;Z$up4toapSossiblefactorassoSciatedwiththeprime2the32'L-seriesis7M@(a;s)UR=Y 0׹x  2!ů( = W) s (a)'where$the$sumistakrenover$alloSddidealsainZ[i]primetoaanda7=6( )'with꨽ hUR1moSd":(2+2i).8Let}P׽aUR=2 v c̻0with3Ƚ=1;UPc̻0V1 moSd&64:(11)'Becaused4isdafourthpSorwerdinQ(i),qwedneedonlyconsideraUR>0. 3Let ܹbSe'theR3crharactersatisfying n9( )UR=(a)R3whereaUR=( )R3andR4 h1moSd":(2sA+sB2i);'andGNextendthisbryGOwriting n9(i)=1.NTheoSddGNpartoff,^xtheconductorof' n9,istheproSductoftheoddprimesdividingaandtheevrenpartisxږ8zif꨽vXisoSddYU;xږ4zif꨽vËUR1+ moSd&64׽;xږ2zif꨽vËUR1 moSd&64andc̻0VUR1 moSd8W޽;xږ1zif꨽vËUR1 moSd&64andc̻0VUR5 moSd8W޽: 18x='ChoSosef)>|*0sothatf=|+(fG).@WVeset n9(1 + i)=i2(1c-:2N0*)=8&iffisoSdd;}the'missingfactorintheL-seriesisthenCaf1(1) (cq0*5)=8"R2 1sJg 1 \|:Cb'Withthisfactorincluded,wredenotetheL-seriesbyL(a;s).8Then\xL(a;2)UR=Fu1z@4 !n9 2.=fG 2yX!Ͻ n9(u)}(!u=fG)]'whereothepsumistakrenoverpasetofrepresentativespuof(Z[i]=(fG))2.6The'mainconclusionof[3]isasfollorws:L 'Theorem1l޺Suppffose8 that8 a^C>^D0andfC>3;:tthena21=4 @ fGf̻2!n922 ʵL(a;2)is8 inZ,'wherffe35f̻29istheevenpartoffG.'The prorviso f>C3isessenrtialhere.Thesubstantial tablesin[3]suggest'that,prorvidedaUR>0andfQ>UR3,Cb2 t .Ba 1=4 @ fG!n9 2 ʵL(a;2)(12)Ca'is^Aa^@pSositivreinteger,{'where^@tisthenrumbSer^Aofodd^@primefactorsoffG.Itis'temptingtoregard!n922.==a21=4 @ fܹasascalingfactor,tocall22tFafudgefactorand'toconjecturethattheinrteger(12)istheorderoftheHV21 h4idenrti edbyBloSch'and GKato.Butinthethreecasesa=1;12and H108forwhicrhf֋<4,(12)is'notaninrteger,sothesituationcannotbSequitesosimple.8As9m9increases,newL-functionsappSear. &BytheCorollarytoLemma'6theyallharvetheformPc(a) 2q1 ݄ 2q2Y( = W)2slwhere̻1;̻2arenon-negativre'inrtegerswith̻1+ ̻2V=URm1.ByanobrviouschangeofvXariables,Gvitisenough'to considerP@˽(a)) 2 )չ( = W)2sԹforsomea,wherem+isanoSddinrteger.The'moreܱinrterestingܲcase,3whichisܱtheonlyonewhicrhwillbSetreatedhere,4is'when+ʽm+ɹisevren;kithecasewhenmisoSddissimilarexceptthattheruleforthe'conductor[ismorecomplicatedanditmarybSenecessarytosupplyafactor'for꨽L(s)correspSondingtotheprime2.8Thruswewrite]L̾3(s)UR=X ҍ a(a))  2+1ڹ( = W) s(13)$f'wherej8UR0isaninrtegerandthesumisoveralloSddaprimetoa.(WVedrop'the[previous\meaningofǹ.)Asusualwretake[ n9( )f=f۽(a)ifa=f( )and' hUR1moSd":(2&s+2i),butthistimewreextendthisde nitionby n9(i)UR=i22+1P. 19x='WVeagainwriteaintheform(11).0PTheoSddpartoff,theconductorof n9,is'theproSductoftheoddprimesdividinga;andthistimetheevrenpartis08ʺif꨽vXisoSdd,4ʺifՀ vËUR1+ moSd&64;q2(1+i)ʺifՀ vËUR1 moSd&64:8ThestrategicvXaluesofsaretheinrtegersfromU+ʍ1to2T+1inclusivre.'TheMtmostMsinrterestingoftheseis+1,f&whicrhisatthecenrtreofthecritical'strip.8TVoconrtinueL̾3(s)intothecriticalstripwewrite%_=0g̾3(z;s)UR=o!nkz 2+1jzj 2s`+X ҍeu 09f(9zݹ+f6u Z) 2+1Pjz3+uj 2s&hqTf6u 2+1juj 2s`(2lo+1s)9zù5u 2!juj 2s+szDqu 2+2juj 22s'4g"6S'wherethesumistakrenoverallnonzerouinthelatticegeneratedby!k'and'i!n9.8Thisseriesisabsolutelyconrvergentinlo+Fuۻ1۟z@2 Q.8InC+Fug3gz@2 xwrecansplitthislastsumintofoursums,eachofwhichis'absolutely conrvergent.Of these,{P(*0J_u22+10ljuj22sXovXanishesidenrticallybSecause'the6terms6cancelinpairs;[andforeacrh߹thesameistrueofoneofthelast'trwosums.8Hencewrehave0 &dIg̾3(z3+!n9;s)g̾(z;s)UR=s!n9h̾+1(s+1)(2lo+1s)!h̾3(s);Ig̾3(z3+i!n9;s)g̾(z;s)UR=si!n9h̾+1(s+1)+(2lo+1s)i!h̾3(s)w'where%wre%havewritten%ٽh̾ (s)=Pd*02u22%juj22sJ.qInparticularthisimpliesthat'h̾3(s)fandfh̾+1(sD+1)canbSefanalyticallyconrtinuedto +Fu2x12xz@2 .Iffis'oSdd,thentriviallyh̾ (s)UR=0.8Norwwrite[g n9ڍ3(z;s)UR=g̾(z;s)+(2lo+1s)9z5h̾(s)szh̾+1(s+1);'so!Rthatg2n9RA3(z;s)!QisadoublypSeriodic!Rfunctionofz,.thoughitisnotanalytic.'Setting꨽sUR=lo+1wrehavesg n9ڍ3(z;lo+1)UR= 5X ㇍jv=0ō7&Y(+1)۟[ziǟ ΍(jW{+1)(jW{+1)սz jjg n9jڍ3(z) I'whereif>129jʦ鴡3(˿)ieܲtʦ0ٙȱ筍(ƙX=UR1@;zZ'X8UX 0;;#u 1Qf(z3+u) 1W.u 1g+h̾3(lo+1)ifj%=UR1)a;'X8UX 0;;#u Gf(z3+u) 1W.u 1+(lo+1)zu 2g(+1)zh̾+1(+2)ifj%=UR0)a:\ 20{x='If5|=s1the rsttrwolinesshouldbSereplacedbryg2n91RA1=}(z).noMoreorverg2n9jRA3(z)'isanevrenfunctionofzsifj{isoSddandanoddfunctionifj{isevren.8Clearly;g n9ڍ˹(z)UR=(1) 1} (1)(z)=(lo+1);'theotherg2n9jRA3(z)aremeromorphicbutnotdoublypSeriodic,andnearz5=UR0bg n9jڍ3(z)UR=h̾(lo+1)+OS(z)for*0UR'is@~meromorphic@}anddoublypSeriodic.(Itsonly@~possiblepoles@}areatthelattice'pSoinrtsUandVitsbeharviournearVz5=UR0canbeVreado fromthatofg2n9jRA3(z).Once'wre+have,determinedthef2GjRA (z)forj%=UR;:::ʜ;˵+˴1thisenablesustodetermine'f2GRA GtupbgtobhthetempSorarilyunknorwnconstantbgC=h߾(1)=2*(lC+{1)when'UR2+:1moSd":4.&ThisconstanrtmustbSecarriedthroughtothenextstep,'atwhicrhwe ndthattheexpressionforf2G1RA(z)conrtainsamultipleofCܞ(z).'Thee~doubleepSeriodicitryofef2G1RA(z)nowdetermineseCܞ.cWhenzbisclosetoa'latticepSoinrt,6caremustbSetakeninevXaluatinggn92 $numericallyV,6bSecausethe'compSonenrtsofitarelargecomparedtotheirsum.8LetFf JbSeEinQ<\<fandnotdivisiblebryanyFprimeinowhicrhdoSesnot'divideSef.sIn(13)Sfwrewrite (=fu!n921+ where 1moSd":(2+2i)runs'throughasetofrepresenrtativesoftherelevrentclassesin(o=f)2.8Thrus#c~-~L̾3(s)UR=qōÑ! ]ݟ[z ΍fq!<2s21G ܟX ҍMI Za2( )g n9ڍ( !n9=f;s)(14)'o'in x+Fu3z@2 i.Bothsides areholomorphicinx+Fu1z@2 wso theconclusion'extendstothatregion. 21&Nx=8SimilarA1argumenrtsgiveclosedformulaeforL̾3(s)forsUR=K+P2;:::ʜ;2L+1;'the.calculations.arenorwsimplersincewredonotneedtoinrtroSduceg̾3(z;s).'InparticularFyiL̾3(2lo+1)UR=ō)o!n922+1[zL 썽fG2+1y(2+1)Tb6X ҍZ g n9( )} () ( !=f):$އ8WVeauillustrateatthisproScessbrycalculatingL̻1(2)andL̻2(3).FFVoratL̻1(2)wre'harve꨽h̻1(s)UR=0andg2n91RA1.=(z)=}(z),whenceI"g n9ڍ1.=(z;2)UR=(9zn9 1 ʵ!n9 2(z))}(z)+f G0ڍ1(z):'But꨽f2G0RA1(z)UR=n921 ʵ!n922.=z23+OS(z)nearz5=UR0,sof2G0RA1(z)UR=Fu33133z@2jn921 ʵ!n922.=}209(z).8ThrusrКg n9ڍ1.=(z;2)UR=(9zn9 1 ʵ!n9 2(z))}(z)Fuۻ1۟z@2 Qn9 1 ʵ!n9 2} 09(z):8Sucrh'formulaeare'convenientforcomputing,7$butnot'forprovingresults'abSoutthenatureofL̾3(s).8FVorthelatterpurpose,wrerecallthat$Ob(z)UR=ōb$rn9z[z ΍.!n92|b2X ҍr0inZ,wherezb=!\;isaninrtegerinQ(i)andthesummandwith'r=UR1isXc} 00r(z)=2} 09(z)UR=(3(}(z)) 2j1)=} 0(z):F'UsingasucienrtlystrengthenedversionoftheKroneckerJugendtraum,it'oughrttobSepossibletoshorwthatifsisastrategicintegerIn9 2+1sH!n9 21Zýa 1=4 @ fGs1Oo1P0b_0t?010?0꿹10ƍ>3P1b_0t?000?1꿹1>9P2b_0t?011?1꿹0827P3b_0t?000?0꿹9>5P0b_1t?010?1꿹1825P0b_2t?000?0꿹12125P0b_3t?000?1꿹9>7P0b_0t?110?1߽16849P0b_0t?201?4꿹02343P0b_0t?310?9꿹4815P1b_1t?010?0꿹9875P1b_2t?000?0꿹42375P1b_3t?040?1߽25845P2b_1t?014?0꿹02225P2b_2t?0025?1꿹0-1125P2b_3t?001?0꿹02135P3b_1t?000?9꿹92675P3b_2t?000?9꿹9-3375P3b_3t?090?0꿹9821P1b_0t?190?1꿹12147P1b_0t?2125?0꿹0-1029P1b_0t?310_25꿹1863P2b_0t?1025?1꿹02441P2b_0t?200?0꿹1-3087P2b_0t?309?4꿹02189P3b_0t?100?9꿹9-1323P3b_0t?2981?9꿹0-9261P3b_0t?300?9߽81835P0b_1t?100?1߽252245P0b_1t?2049?0꿹0-1715P0b_1t?300_25꿹9 24Okx=p5D175b0s214ҿ0071>1225b0s221߽16970>8575b0s234ҿ001?81D875b0s310ҿ0071>6125b0s329߽25070842875b0s330ҿ001?810ƍD105b1s110ҿ001?81D735b1s120ҿ4970>5145b1s130ҿ001?25D525b1s210ҿ0251?81>3675b1s229߽252570825725b1s230ҿ04+_361>2625b1s31100ҿ0079818375b1s320ҿ40702128625b1s33676ҿ00+_169D315b2s119ҿ42570>2205b2s120ҿ0471815435b2s1325169170>1575b2s210100070811025b2s2225ҿ001?81877175b2s230121070>7875b2s311100070855125b2s3264ҿ0?100+_1692385875b2s331߽49070D945b3s1136ҿ00+_225>6615b3s120ҿ9070846305b3s1336ҿ0079>4725b3s219ҿ001?81833075b3s229߽360702231525b3s23144ҿ09+_441823625b3s310ҿ00792165375b3s3202253670-1157625b3s330ҿ00%1089 25Xx=8WVrite{N=L1for{thesigninthefunctionalequation.Ifthesignis1'then L̻1(2)D=0;<>:|Zn̻2=0and|c̻23;9;11orG13 moSd&616;faZn̻2=1;Zn̻2=2and|c̻21;9;11orG15 moSd&616;,n卑'and꨽̻2V=UR1otherwise.8FVorpoSdd,8̾p=UR1ifn̾p=2and|p3 moSd&64'and꨽̾p=UR1otherwise.8Thisisprobablynothardtoprorve,thoughtedious.8The follorwing smalltable,,which coversthosea divisibleonlyby 2,3and'5,givresthevXaluesofw2 32tJf G2ڍo0P0qC0w?zꕹ0000)000݊k4jg濽000t8TQ0000ƍ-3>1P0qC2w?zꕹ0000)000תo16jg濽000100TQ000-9>2P0qC0w?zꕹ00036)000݊k0jg濽000t0TQ000 27oàx=EI$]?827P3b_0}u;1853000?0u000*_0 [_000!36-00000ƍ>5P0b_1}u;1653000?0u000*_0 [_000'4-0000825P0b_2}u;3253000?0u000*_4 [_000ߡ324-00002125P0b_3U7053000?0u000*_0 [_000ߡ196-0000815P1b_1U7053000?0u000Jc32 [_000ߡ338-0000875P1b_2w?12153000?0u000*_0 [_000!64-00002375P1b_3U7053000?0u000*_0 [_000ߡ722-0000845P2b_1}u;7253000576u000jg162 [_000'0-00002225P2b_2w?28853000_18u000*_0 [_000'0-0000-1125P2b_3U7053000900u000Jc18 [_000'0-00002135P3b_1U7053000?0u000*_0 [_000!18-00002675P3b_2w?22553000?0u000Jc72 [_000ߡ324-0000-3375P3b_3U7053000?0u000jg288 [_0005202-0000mx8ThesignofthefunctionalequationisagaintheproSductoflocalfactors,'butthistimetherulesare)Z@̻2V=UR1if̼8 ԍ><>:|Zn̻2=0and|c̻21;7;13orG15 moSd&616;faZn̻2=1;Zn̻2=2and|c̻25;7;11orG13 moSd&616;,!K'and꨽̻2V=UR1otherwise.8FVorpoSdd,2f8̾p=UR1ifn̾p=2and|p3 moSd&64'and꨽̾p=UR1otherwise.986.8The35cffasesnUR=535andnUR=10.[Notyretwritten]'PREFERENCES!$'[1]LBerndt,B.C.,l>EvXans,R.J.andWilliams,K.S.,GaussandJacobiSums(Wi-'leyV,1998).iԍ'[2]Bircrh,B.J.andSwinnerton-Dyer,H.PV.F.,NotesonellipticcurvesISI,J.fvur'reineangew.8Math.218(1965),79-108.'[3]ZPincrh,R.G.E.andSwinnerton-Dyer,H.PV.F.,Arithmeticofdiagonalquartic'surfacesk}I,k~inL-functionsandArithmetic(ed. _J.CoatesandM.J.TVarylor),'LondonMath.8SoSc.LectureNoteseries153(Camrbridge,1991). 28x='[4] ~ShioSda,T.,4TheHodgeConjectureforFVermatVarieties,3Math. bAnn.'245(1979),175-184.'[5]&Swinnerton-Dyrer,H.PV.F.,TheConjecturesofBirchandSwinnerton-Dyer,'andpofTVate,HinProSceedingsofaConferenceonLocalFields(ed.xT.A.Springer)'(Springer,1967).'[6]IWVashington,L.C.,InrtroSductiontoCyclotomicFields(2ndJed.,Springer,'1991).'[7]WVeil,A., Jacobisumsas"Grossencrharaktere",TVrans.Amer.Math.SoSc.'73(1952),487-495=OeuvresScienrti quesISI,63-71.'[8]WVeil,A.,ISommesdeJacobietcrharactseresdeHecrke,HG ott.8NNachr.(1974),'Nr1=OeuvresScienrti quesISII,329-342. 29;x.DF cmmib10%\%eufm8$%n eufm10"2@cmbx8!N cmbx12 msam10u cmex10q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8@ cmti12XQ cmr12G