; TeX output 2004.06.16:1807 pK5ܥcolor push Black color popbˍ- color push Black color popxDt q G cmr17AnBExistenuNtialDivisibilityLemmaforGlobalFields/_썍xXQ ff cmr12JerodCen/Demeyer= !", cmsy10 Jan/V4anGeel=y'. Ǡ»2004{06{16RuBܥN G cmbx121Inutro =duction&"ܥXQ cmr12Inx[Phe00~]ThanasesPheidasprorves,whatxhecalls,existenrtialdivisibilitylemmasfor"g cmmi12K$= ܥ. msbm10F#2 cmmi8q(z )>whereqisaprimecongruenrtto3URmoSd/4andforK3=Q.5`LetRXbSethesetofprimesܥp6inZ,respSectivrelyinFq[t],sucrhthat%!", cmsy10 1isnotasquareintheresidueeldofp. Theܥexistenrtialfdivisibilitylemmasstatetheexistenceofanexistentialformula(x;yn9)suchthat:ܥ1@ cmti12if-4(x;yn9)holdsforx;yË2URKynPf0gthenforal lprimesp2RF~forwhichvp](x)isoffdditfol lowsܥthat35vp](xyn92&K cmsy8 |{Y cmr82ʵ)UR>0.?ܥActually!puttingextraloScalconditionsontheelemenrtsx(conditionsinthepoinrtatinnityܥinthefunctioneldcaseandintherealprimeandtheprime2incaseK1=URQ)thetruthofܥ(x;yn9)+forx;y 2bK ܞ2 isequivXalenrttothestatement:for[4al lprimespb2Rt~for[4whichvp](x)isܥoffdd35itfol lowsthatvp](xyn92 2ʵ)UR>0.ܥTheformrula(x;yn9)expresses\almost"thatal lpffolesofxinRJ,withoddmultiplicity,areܥpffoles35ofyn9.ܥTheseexistenrtialdivisibilitylemmasplayaroleinstrategiestoobtainundecidabilityresultsܥfor theexistenrtialtheoryoftheeldK ܞ.In[Phe91~]PheidasprovedthattheexistentialtheoryܥofkFp$; cmmi6n 唹(t)isundecidable.CHisproSofwrorkedkforalloddprimesp.CVidela([Vid94Y])extendedܥthisresulttorationalglobalfunctioneldsofcrharacteristic2.ShlapSentokh([Shl96])showedܥ color push Blackff ff B g^O! cmsy7 K`y cmr10TheDrstauthorisaResearchAssistantoftheF*undforScienticResearch-Flanders(Belgium)(F.W.O.- Vlaanderen). X^y W*orkpartiallysuppGortedbytheEuropeanCommunity'sHumanPotentialProgrammeundercontractHPRN-CT-2002-00287. color pop ܥcolor push Black 1 color pop *pK5ܥcolor push Black color popbˍK5ܥthattheexistenrtialtheoryofglobalfunctioneldsofcharacteristicnot2isundecidableand ܥnallyZinherthesis([Eis03])KirstenEisenrtr agercompletedtheresultsbyprovingthesameܥresultڨforglobalfunctioneldsincrharacteristic2.3Thisisessentiallythe(negative)solutionܥofHilbSert's10thproblemforglobalfunctionelds.[ܥOneuofthemainopSenquestionsdirectlyrelatedtoHilbert's10thproblemiswhetherornotܥthewexistenrtialtheoryofQ(ormoregeneralofanynumbSereld)isdecidableorundecidable.ܥIn1[Phe00~]aprogramtocometoauniformwray1toattacrkHilbSert's10thproblemforglobalܥeldseisdescribSed.(Thisprogramgeneratesaseriesof\possiblefacts"(cf.[Phe00~,sectionsܥ3(and4]),xtrwo(ofwhicrharerelatedtotheexistentialdivisibilitylemmaandoneofthemܥis^forcedinrtothestrategybythefactthattheexistentialdivisibilitylemmaasprovedbyܥPheidasusesthesetofprimespforwhicrh 1isnotasquareintheresidueeldofp.ܥIn1viewofthisThanasesPheidasandGunrtherCornelissenraisedthequestiontowhatextentܥtheUexistenrtialdivisibilitylemmaholds.Forwhichsetsofprimesdoffesithold>?Whydoffestheܥcffondition35intheprime2occurJ?8Canthelemmabegeneralizedtoal lglobalelds ݹ?ܥTVogether%withKarimZahidithesecondauthorwrorked%outamoregeneralvrersionoftheܥexistenrtialdivisibilitylemma.uRNamelyforKT<=wQandforasetofprimesthatareinertinaܥquadraticmextensionofQ.ThisresultwraspresentedattheObSerwolfachmeetingonHilbSert'sܥ10thprobleminJanruary2003(cf.[VZ03]).ܥInq!thispapSerwregeneralizetheexistentialdivisibilitylemmatoallglobaleldsKM(ofcharac-ܥteristicrnot2),andforallsetsofprimesthatareinertinaquadraticextensionLofK ܞ.V=WVeܥrstLgprorvethedirectgeneralisationofPheidas'existentialdivisibilitylemma,lwithconditionsܥinltherealprimesandintheprimesramifyinginL.+Inthelastsectionwreremovealltheseܥconditionsandprorve:ܥLffetKfbeaglobaleldandRJ(L=K ܞ)bethesetofprimeswhichareinertinaquadraticܥextensiony9LofK ܞ.8uThentherffeisanexistentialformula (x;yn9)whichisequivalentwiththeܥformulaz82%n eufm10rUR2RJ(L=K ܞ):G(u cmex10 Tv3\% eufm8r>(x)35offdd"+!vr(xyn9 2ʵ)>0G14ܥ2Preliminaries$=8ܥOur discussionoftheexistenrtialdivisibilitylemmareliesonfactsabSoutnormsandnormܥgroupsuLofquadraticextensionsofloScalandglobalelds.WVeusetheHasse{MinkrowskiuLlocal{ܥglobal principleandHilbSert'sreciprocitrylaw.-WInthissectionwegiveasurveyofthesefacts,ܥformoredetailsandproSofswrerefertotheliterature(e.g.[O'M63"I]). ܥcolor push Black 2 color pop BpK5ܥcolor push Black color popbˍK5ܥWVe@startxingterminologyandnotation.pThroughoutthepapSerKwillbeaglobaleldof ܥcrharacteristicbnot2, soitiseitheranumbSereldorthefunctioneldofacurveoveraniteܥeldFqwithqXoSdd.8LwillbeaquadraticextensionofK ܞ.QܥWith*xMKf:wredenotethesetofall\primes"pofK ܞ.InthenumbSereldcasethenite(ornon-ܥarffchimedeanSLprimesmcorrespSond(onetoone)to(equivXalenceclassesof 8)discretevaluationsofܥKandtheinniteȹorarffchimedeanprimescorrespSond(onetoone)tothedierenrtembSeddingsܥofKKinthecomplexnrumbSers.InthefunctioneldcasealltheelemenrtsofMKUocorrespondܥ(onetoone)to(equivXalenceclassesof 8)discretevaluationsonK ܞ. SWithevreryelementofMKܥtherecorrespSondsanormalizedabsolutevXaluej*jpIonK ܞ,wreletKpdenotethecompletionofܥKFwithrespSecttothisabsolutevXalue.ܥIfcpisanon-arcrhimedeanprime,~,thenKpisthefractioneldofacompletediscretevXaluationܥringNOp anditsmaximalidealisaprincipalideal. dmAN&generatorforthisidealiscalledaܥuniformizingelemenrt.4WVecanchoSosesuchauniformizingelementinthebaseeldK`andܥdenoteqitwithp.ThequotienrtringOp=(p)isaniteeldFp,theresidueeldoftheprimeܥp.The^discretevXaluationassoSciatedwithanon-arcrhimedeanprimepwillbedenotedwithܥvp and~wrenormalizeitbyvp(p)Qm=1.6WVe~usetheanalogousterminologyandnotationforܥprimesPUR2MLGع.ܥAllRabsolutevXaluesj*jpYextendtothequadraticextensionL=Kindierenrtways,depSendingܥonVtheprimep.}ThepSossibleextensionscorrespondtowhatiscalledthesplittingbffehaviorܥoftheprimepinL.)FVoraquadraticextensionwrehavethefollowingpSossibilities(intheܥnrumbSer8/eldcasewredistinguishbetrween8/non-archimedeanandarchimedeanprimes,KintheܥfunctioneldcasepSoinrts4and5areempty):##ܥcolor push Black\h1. color pop