Pure L-functions from algebraic geometry over fini

PureL-FunctionsfromAlgebraicGeometryoverFiniteFieldsDaqingWanDepartmentofMathematics,UniversityofCalifornia,Irvine,[e-mail]andintuitivelyself-containedintroductiontothetheoryofpureL-functionsarisingfromafamilyofalgebraicvarietiesdenedoveraniteeldofcharacteristicp.ThestandardfundamentalquestionsinanytheoryofL-functionsincludethemeromorphiccontinuation,functionalequation,Riemannhypothesis(RHforshort),orderofzerosatspecialpointsandtheirspecialvalues.OuremphasisherewillbeonthemeromorphiccontinuationandtheRH.Thesetwoquestionscanbedescribedinageneralsetupwithoutintroducinghighlytechnicalterms.TheconstructionofthepureL-functiondependsonthechoiceofanabsolutevalueoftherationalnumbereldQ.Inthecasethattheabsolutevalueisthecomplexor‘-adicabsolutevalue(‘6=p),therationalityofthepureL-functionrequiresthefullstrengthofthe‘-adiccohomologyincludingDeligne’smaintheoremontheWeilconjectures.Inthecasethattheabsolutevalueisthep-adicabsolutevalue,thepureL-functionisnolongerrationalbutconjecturedbyDworktobep-adicmeromorphic.Thisconjecturegoesbeyondallexistingp-adiccohomologytheories.Itstruthopensupseveralnewdirectionsincludingapossiblep-adicRHforsuchpureL-functions.Theguidingprincipleofourexpositioninthispaperistodescribealltheoremsandproblemsassimpleaspossible,directlyintermsofzetafunctionsandL-functionswithoutusingcohomologicalterms.Inthecasethatthisisnoteasytodoso,wesimplygiveanintuitivediscussionandtrytoconveyalittlefeeling.Alongtheway,anumberofnaturalopenquestionsandconjecturesareraised,someofthemmaybeaccessibletocertainextentbutothersmaybesomewhatwildduetothelackofsucientevidences.1IntroductionThemostbasicquestioninnumbertheoryistounderstandtheintegers.Inparticular,foragivenintegerN,weneedtounderstandtheabsolutevaluekNkforeveryabsolutevaluek?kontherationalnumbereldQ.Forthecomplexabsolutevalue,thisistodeterminekNk=jNj=?Forthep-adicabsolutevaluewithpbeingaprime,thisistodeterminejNjp=pap;ap=ordp(N)=?Thislasttheoreticalquestionispracticallytheproblemoffactoringintegerswhichhasimportantapplications.Moregenerally,supposethatwearegivenasequenceofinterestinginte-gersfN1;N2;;gInordertounderstandthissequenceofintegers,onenaturallyformsasuit-ablegeneratingfunctionZ(fNig;T)whichcontainsallinformationaboutthegivensequence.Thebasicquestionisthentounderstandtheanalyticproper-tiesofthegeneratingfunctionZ(fNig;T)withrespecttoeachabsolutevaluek?kofQ.ThisincludesthepossiblemeromorphiccontinuationZ(fNig;T)andasuitableRHaboutitszerosandpoles,forboththecomplexabsolutevalueandthep-adicabsolutevalue.Ifwehaveafamilyofsuchgeneratingfunctions,thenwewouldliketounderstanditsanalyticvariationwhentheparametervaries.Themostinterestingtypeofsequencesarisesfromcountingprimeidealsinanitelygeneratedcommutativeringorequivalentlyfromcountingrationalpointsonanalgebraicvariety.InthecaseofcountingprimenumbersintheringZofintegers,thenaturalgeneratingfunctionistheRiemannzetafunction.ThisrstexamplewasstudiedbyRiemannfromcomplexpointofviewandbyKummer-Kubota-Leopoldtfromp-adicpointofview.ItisthemotivatingexampleformuchofthemoderndevelopmentsongeneralHasse-Weilzetafunctionsofalgebraicvarietiesaswellastheirconjecturalp-adicanalogues.OurinterestingsequenceofintegersinthispaperarisesfromcountingrationalpointsovervariousniteextensioneldsofanalgebraicvarietyXdenedoveraniteeldofcharacteristicp.TheresultinggeneratingfunctionisthezetafunctionofXwhichistheobjectofstudyinthecelebratedWeilconjectures.ThezetafunctionisarationalfunctionasprovedbyDworkusingp-adicmethods.Itsatisesasuitablecomplexand‘-adicRHasprovedbyDeligneusing‘-adicmethods,where‘isaprimenumberdierentfromp.Thep-adicRHforthezetafunctionismorecomplicatedandremainsmysteriousingeneral.Thevariationofthewholezetafunction,whenthevarietymovesthroughanalgebraicfamily,leadstonewinterestingquestionswhichareunderstoodtocertainextent.Thezetafunctionishowevernotpure.Thatis,thezerosandpoleshavedierentabsolutevalues.Thisisespeciallysofromp-adicpointofview.Thus,thezetafunctiondecomposesasaproductofpurepiecesdenedintermsoftheabsolutevaluesofthezerosandpoles.AnerformoftheRHistounderstandthepuritydecomposition.Afurtherquestionistounder-standthevariationofeachpurepieceofthezetafunctionwhenthevarietymovesthroughanalgebraicfamily.ThisnaturallyleadstotheconstructionofpureL-functionsarisingfromalgebraicgeometry.OurfundamentalquestionhereisthentounderstandtheanalyticpropertiesofsuchapureL-function,notablyitsmeromorphiccontinuationandRH.Sincethezetafunctionhasintegercoecients,therearethreedierenttypesofabsolutevalues(com-plex,‘-adicandp-adic)thatwecanchoosetowork.Theseleadtodierentresultsanddierenttheories.Inthecasethattheabsolutevalueisthecomplexorthe‘-adicabsolutevalue,Deligne’smaintheoremshowsthatthepureL-functionfromalgebraicgeometrycanbeidentiedwithageometricL-function,thatistheL-functionofacertaingeometricconstructible‘-adicetalesheaf.Onecanthenapplythefullmachineryof‘-adicetalecohomology.Inparticular,thepureL-functionfromalgebraicgeometryisalwaysrationalbyG