Trace formula in noncommutative geometry and the z

arXiv:math/9811068v1[math.NT]10Nov1998TraceformulainnoncommutativeGeometryandthezerosoftheRiemannzetafunctionAlainCONNESAbstractWegiveaspectralinterpretationofthecriticalzerosoftheRie-mannzetafunctionasanabsorptionspectrum,whileeventualnoncriticalze-rosappearasresonances.WegiveageometricinterpretationoftheexplicitformulasofnumbertheoryasatraceformulaonthenoncommutativespaceofAdeleclasses.ThisreducestheRiemannhypothesistothevalidityofthetraceformulaandeliminatestheparameterδofourpreviousapproach.TableofcontentsIntroduction.IQuantumchaosandthehypotheticalRiemannflow.IIAlgebraicGeometryandglobalfieldsofnonzerocharacteristic.IIISpectralinterpretationofcriticalzeros.IVThedistributiontraceformulaforflowsonmanifolds.VTheaction(λ,x)→λxofK∗onalocalfieldK.VITheglobalcase,andtheformaltracecomputation.VIIProofofthetraceformulaintheS-localcase.VIIIThetraceformulaintheglobalcase,andeliminationofδ.AppendixI,Proofoftheorem1.AppendixII,Explicitformulas.AppendixIII,Distributiontraceformulas.1IntroductionWeshallgiveinthispaperaspectralinterpretationofthezerosoftheRiemannzetafunctionandageometricframeworktowhichonecantransposetheideasofalgebraicgeometryinvolvingtheactionoftheFrobeniusandtheLefchetzformula.Thespectralinterpretationofthezerosofzetawillbeasanabsorptionspectrum,i.e.asmissingspectrallines.Allzeroswillplayaroleinthespectralsideofthetraceformula,butwhilethecriticalzeroswillappearper-se,thenoncriticaloneswillappearasresonancesandenterinthetraceformulathroughtheirharmonicpotentialwithrespecttothecriticalline.Thusthespectralsideisentirelycanonical,andbyprovingpositivityoftheWeildistribution,weshallshowthatitsequalitywiththegeometricside,i.e.theglobaltraceformula,isequivalenttotheRiemannHypothesisforallL-functionswithGr¨ossencharakter.WeshallmodelourdiscussionontheSelbergtraceformula,butitdiffersfromthelatterinseveralimportantrespects.Weshallfirstexplaininparticularwhyacrucialnegativesignintheanalysisofthestatisticalfluctuationsofthezerosofzetaindicatesthatthespectralinterpretationshouldbeasanabsorptionspectrum,orequivalentlyshouldbeofacohomologicalnature.Asitturnsout,thegeometricframeworkinvolvesaninnocentlookingspace,thespaceXofAdeleclasses,wheretwoadeleswhichbelongtothesameorbitoftheactionofGL1(k)(kaglobalfield),areconsideredequivalent.ThegroupCk=GL1(A)/GL1(k)ofIdeleclasses(whichistheclassfieldtheorycounterpartoftheGaloisgroup)actsbymultiplicationonX.Ourfirstpreliminaryresult(theorem1ofsectionIII)givesaspectralinterpretationofthecriticalzerosofzetaandLfunctionsonaglobalfieldkfromtheactionoftheIdeleclassgrouponaspaceofsquareintegrablefunctionsonthespaceX=A/k∗ofAdeleclasses.Corollary2givesthecor-respondingcomputationofthespectraltrace.ThisresultisonlypreliminarybecauseitrequirestheuseofanunnaturalparameterδwhichplaystheroleofaSobolevexponentandallowstoseetheabsorptionspectrumasapointspectrum.Oursecondpreliminaryresultisaformalcomputation(sectionVI)ofthecharacteroftherepresentationoftheIdeleclassgroupontheaboveL2space.ThisformalcomputationgivestheWeildistributionwhichistheessentialingredientoftheRiemann-Weilexplicitformula.Atthispoint(whichwasthesituationin[Co]),themainproblemsaretogivearigorousmeaningtotheformaltracecomputationandtoeliminatetheunwantedparameterδ.2Thesetwoproblemswillbesolvedinthepresentpaper.Wefirstproveatraceformula(theorem3ofsectionV)fortheactionofthemultiplicativegroupK∗ofalocalfieldKontheHilbertspaceL2(K),and(theorem4ofsectionVII)atraceformulafortheactionofthemultiplicativegroupCSofIdeleclassesassociatedtoafinitesetSofplacesofaglobalfieldk,ontheHilbertspaceofsquareintegrablefunctionsL2(XS),whereXSisthequotientofQv∈SkvbytheactionofthegroupO∗SofS-unitsofk.InbothcasesweobtainexactlythetermsoftheWeilexplicitformulaswhichbelongtothefinitesetofplaces.ThisresultisquiteimportantsincethespaceXSishighlynontrivialassoonasthecardinalityofSislargerorequalto3.IndeedthisquotientspaceisnontypeIinthesenseonNoncommutativeGeometryanditisreassuringthatthetraceformulacontinuestoholdthere.Wecheckindetail(theorem6ofAppendixII)thattherewritingoftheWeilexplicitformulaswhichispredictedbytheglobaltraceformulaiscorrect.Finally,weeliminateinsectionVIII,usingideaswhicharecommonbothtotheSelbergtraceformulaandtothestandardexplanationoftheabsorptionlinesinphysics,theunpleasantparameterδwhichappearedasalabelofthefunctionspacesofsectionIII.WewritetheglobaltraceformulaasananalogueoftheSelbergtraceformula.Thevalidityofthetraceformulaforanyfinitesetofplacesfollowsfromtheorem4ofsectionVII,butintheglobalcaseisleftopenandshown(Theorem5ofsectionVIII)tobeequivalenttothevalidityoftheRiemannHypothesisforallLfunctionswithGr¨ossencharakter.Thisequivalence,togetherwiththeplausibilityofadirectproofofthetraceformulaalongthelinesoftheorem4(sectionVII)constitutethemainresultofthispaper.Theeliminationoftheparameterδisthemainimprovementofthepresentpaperwithrespectto[Co].Itisanoldidea,duetoPolyaandHilbertthatinordertounderstandthelocationofthezerosoftheRiemannzetafunction,oneshouldfindaHilbertspaceHandanoperatorDinHwhosespectrumisgivenbythenontrivialzerosofthezetafunctio