A Riemann hypothesis for characteristic p L-functi

ARIEMANNHYPOTHESISFORCHARACTERISTICpL-FUNCTIONSDAVIDGOSSAbstract.WeproposeanalogsoftheclassicalGeneralizedRiemannHypothesisandtheGeneralizedSimplicityConjectureforthecharacteristicpL-seriesassociatedtofunctioneldsoveraniteeld.Theseanalogsarebasedontheuseofabsolutevalues.Furtherweuseabsolutevaluestogivesimilarreformulationsoftheclassicalconjectures(with,perhaps,nitelymanyexceptionalzeroes).Weshowhowbothsetsofconjecturesbehaveinremarkablysimilarways.1.IntroductionThearithmeticoffunctioneldsattemptstocreateamodelofclassicalarithmeticusingDrinfeldmodulesandrelatedconstructionssuchasshtuka,A-modules,-sheaves,etc.LetkbeonesuchfunctioneldoveraniteeldFrandlet1beaxedplaceofkwithcompletionK=k1.ItiswellknownthatthealgebraicclosureofKisinnitedimensionaloverKandthat,moreover,Kmayhaveinnitelymanydistinctextensionsofaboundeddegree.Thusfunctioneldsareinherently\looserthannumbereldswherethefactthat[C:R]=2oersconsiderablerestraint.Assuch,objectsofclassicalnumbertheorymayhavemanydierentfunctioneldanalogs.Classifyingthedierentaspectsoffunctioneldarithmeticisalengthyjob.OnendsforinstancethattherearetwodistinctanalogsofclassicalL-series.OneanalogcomesfromtheL-seriesofDrinfeldmodulesetc.,andistheoneofinteresthere.TheotheranalogarisesfromtheL-seriesofmodularformsontheDrinfeldrigidspaces,(see,forinstance,[Go2]).Itisaverycuriousphenomenonthattherstanalogpossessesnoobviousfunctionalequationwhereasthesecondoneindeedhasafunctionalequationverysimilartotheclassicalversions.ItisevenmorecuriousthattheL-seriesofDrinfeldmodulesandthelikeseemtopossessthecorrectanalogsoftheGeneralizedRiemannHypothesisandtheGeneralizedSimplicityConjecture(seeConjecture3below).Itisourpurposeheretodenethesecharacteristicpconjecturesandshowjusthowclosetheyaretotheirclassicalbrethren.ThattheremightbeagoodRiemannHypothesisinthecharacteristicptheoryrstarosefromtheground-breakingwork[W1]ofDaqingWan.Inthispaper,andinthesimplestpossiblecase,WancomputedthevaluationsofzeroesofananalogoftheRiemannzetafunctionviathetechniqueofNewtonpolygons.Thisimmediatelyimpliedthatthesezeroesareallsimpleandlieona\line.However,becauseofthegreatsizeofthefunctioneldarena(asmentionedabove),itwasnotimmediatelyclearhowtothengoontostateaRiemannHypothesisinthefunctioneldcasewhichworkedforallplacesofk(asexplainedinthispaper)andallfunctionsarisingfromarithmetic.Recently,theL-functionsoffunctioneldarithmeticwereanalyticallycontinuedintotalgenerality(asgeneralasonecouldimaginefromtheanalogywithclassicalmotives).ThisisDate:October,1999.ThispaperisrespectfullydedicatedtoBernardAlterandShirleyHasnas.12DAVIDGOSSduetotheforthcomingworkofG.BoeckleandR.Pink[BP1]whereanappropriatecoho-mologytheoryiscreated.Thistheory,combinedwithcertainestimatesprovidedbyBoeckle,ontheonehand,andY.Amice[Am1],ontheother,actuallyallowsonetoanalyticallycon-tinuethenon-ArchimedeanmeasuresassociatedtotheL-series;theanalyticcontinuationoftheL-seriesthemselvesthenarisesasacorollary.InparticularwededucethatallsuchL-functions,viewedatallplacesofk,haveremarkablysimilaranalyticproperties(forinstance,theirexpansioncoecientsalldecayexponentially|seethediscussionafterRemarks2).Motivatedbytheseresults,were-examinedtheworkofWanandthosewhocameafterhim([DV1],[Sh1]).InseekingtorephraseWan’sresultsinsuchawayastoavoidhavingtocomputeNewtonpolygons(whichlookstobeexceedinglycomplicatedingeneral),wearrivedatastatementinvolvingonlytheuseofabsolutevaluesofzeroes(asopposedtotheabsolutevaluesofexpansioncoecientswhichareusedinNewtonpolygons).TheuseofabsolutevaluesinphrasingsuchapossibleRiemannHypothesisseemstobeveryfruitful.Forinstance,itoersaunicationwithlocalRiemannHypotheses(whicharealwaysformu-latedintermsofabsolutevaluesofthezeroes).Morestrikingly,italsosuggestsasuitablereformulationoftheclassicalGRH(with,perhaps,nitelymanyexceptionalzeroes)aswellasthesimplicityconjectures(seeConjecture6andProposition7).Finally,asexplainedafterRemarks4,theconjecturespresentedheregoaverylongwaytowardsexplainingthelackofaclassicalstylefunctionalequationassociatedtotheL-seriesofDrinfeldmodulesetc.Uponexaminingthesenew\absolutevalueconjecturesinboththeories,onendsthattheybehaveremarkablyalike.SomuchsothattheyseemtoalmostbetwoinstancesofonePlatonicmold.Thiscertainlyaddstooursensethatthefunctioneldstatementsmayindeedbethecorrectones.Moreover,becausethealgebraicclosureofKissovastandcontainsinseparableextensions,thefunctioneldtheoryoersinsightintothesestatementsnotavailableinnumberelds.Forinstance,duetotheexistenceofinseparableextensions,oneneedsboththefunctioneldanalogoftheGRHandthefunctioneldanalogoftheSimplicityConjecture(Conjecture7)totrulydeducethatthezeroes(oralmostallofthem)lieonaline!BecauseCisobviouslyseparableoverRoneonlyneedstheGRH,(reformulatedasConjecture6)classically.Itshouldbenotedthatwedonotyetknowtheimplicationsofourfunctioneldcon-jectures.However,itisourhopethatsuchinformationwillbefoundasabyproductofthesearchforaproofofthem.Moreover,becauseofthestrongformalanalogiesbetweenthenumbereldandfunctioneldconjecturesasprese