1 Introduction Detailed Research Plan

  • 格式:pdf
  • 大小:96.71 KB
  • 文档页数:7

DetailedResearchPlan1IntroductionMyresearchprogramreflectstheessentialinterplaybetweenabstracttheoryandexplicitmachinecomputationduringthelatterhalfofthetwentiethcentury;itsitsattheintersec-tionofrecentworkofB.Mazur,K.Ribet,J-P.Serre,R.Taylor,andA.WilesonGaloisrepresentationsattachedtomodularabelianvarieties(see[21,24,26,28])withworkofJ.Cremona,N.Elkies,andJ.-F.Mestreonexplicitcomputationsinvolvingmodularforms(see[9,11]).In1969B.Birch[4]describedcomputationsthatledtothemostfundamentalopenconjectureinthetheoryofellipticcurves:

IwanttodescribesomecomputationsundertakenbymyselfandSwinnerton-DyeronEDSACbywhichwehavecalculatedthezeta-functionsofcertainellipticcurves.AsaresultofthesecomputationswehavefoundananalogueforanellipticcurveoftheTamagawanumberofanalgebraicgroup;andconjectures(duetoourselves,duetoTate,andduetoothers)haveproliferated.

Therichtapestryofarithmeticconjecturesandtheoryweenjoytodaywouldnotexistwith-outtheground-breakingapplicationofcomputingbyBirchandSwinnerton-Dyer.Com-putationsinthe1980sbyMestrewerekeyinconvincingSerrethathisconjecturesonmodularityofoddirreducibleGaloisrepresentationswereworthyofseriousconsideration(see[24]).Theseconjectureshaveinspiredmuchrecentwork;forexample,Ribet’sproofofthe󰀜-conjecture,whichplayedanessentialroleinWiles’sproofofFermat’sLastTheorem.MyworkontheBirchandSwinnerton-DyerconjectureformodularabelianvarietiesandsearchfornewexamplesofmodularicosahedralGaloisrepresentationshasledmetodiscoverandimplementalgorithmsforexplicitlycomputingwithmodularforms.Myresearch,whichinvolvesfindingwaystocomputewithmodularformsandmodularabelianvarieties,isdrivenbyoutstandingconjecturesinnumbertheory.

2InvariantsofmodularabelianvarietiesNowthattheShimura-Taniyamaconjecturehasbeenproved,themainoutstandingprob-leminthefieldistheBirchandSwinnerton-Dyerconjecture(BSDconjecture),whichtiestogetherthearithmeticinvariantsofanellipticcurve.ThereisnogeneralclassofellipticcurvesforwhichthefullBSDconjectureisknown.ApproachestotheBSDconjecturethatrelyoncongruencesbetweenmodularformsarelikelytorequireadeeperunderstandingoftheanalogousconjectureforhigher-dimensionalabelianvarieties.Asafirststep,Ihaveob-tainedtheoremsthatmakepossibleexplicitcomputationofsomeofthearithmeticinvariantsofmodularabelianvarieties.

2.1TheBSDconjectureBy[6]wenowknowthateveryellipticcurveoverQisaquotientofthecurveX0(N)whosecomplexpointsaretheisomorphismclassesofpairsconsistingofa(generalized)ellipticcurveandacyclicsubgroupoforderN.LetJ0(N)denotetheJacobianofX0(N);thisisanabelianvarietyofdimensionequaltothegenusofX0(N)whosepointscorrespondtothedegree0divisorclassesonX0(N).AnoptimalquotientofJ0(N)isaquotientbyanabeliansubvariety.ConsideranoptimalquotientAsuchthatL(A,1)=0.By[13],A(Q)andX(A/Q)arebothfinite.TheBSDconjectureassertsthatL(A,1)ΩA=#X(A/Q)·󰀁p|Ncp#A(Q)·#A∨(Q).

HeretheShafarevich-TategroupX(A/Q)isameasureofthefailureofthelocal-to-globalprinciple;theTamagawanumberscparetheordersofthecomponentgroupsofA;therealnumberΩAisthevolumeofA(R)withrespecttoabasisofdifferentialshavingeverywherenonzerogoodreduction;andA∨isthedualofA.Mygoalistoverifythefullconjectureformanyspecificabelianvarietiesonacase-by-casebasis.ThisisthefirststepinaprogramtoverifytheaboveconjectureforaninfinitefamilyofquotientsofJ0(N).

2.2TheratioL(A,1)/ΩA

FollowingY.Manin’sworkonellipticcurves,A.Agash´eandIprovedthefollowingtheoremin[2].

Theorem1.LetmbethelargestsquaredividingN.TheratioL(A,1)/ΩAisarationalnumberthatcanbeexplicitlycomputed,uptoaunit(conjecturally1)inZ[1/(2m)].

TheproofusesmodularsymbolscombinedwithanextensionoftheargumentusedbyMazurin[17]toboundtheManinconstant.TheratioL(A,1)/ΩAisexpressedasthelatticeindexoftwomodulesovertheHeckealgebra.Iexpectthemethodtogivesimilarresultsforspecialvaluesoftwists,andofL-functionsattachedtoeigenformsofhigherweight.IhavecomputedL(A,1)/ΩAforalloptimalquotientsoflevelN≤1500;thistablecontinuestobeofvaluetonumbertheorists.

2.3ThetorsionsubgroupIcancomputeupperandlowerboundson#A(Q)tor,butIcannotdetermine#A(Q)tor

inallcases.Experimentally,thedeviationbetweentheupperandlowerboundisreflected

incongruenceswithformsoflowerlevel;Ihopetoexploitthisinapreciseway.Ialsoobtainedthefollowingintriguingcorollarythatsuggestscancellationbetweentorsionandcp;itgeneralizestohigherweightforms,thussuggestingageometricexplanationforreducibilityofGaloisrepresentations.

Corollary2.Letnbetheorderoftheimageof(0)−(∞)inA(Q),andletmbethelargestsquaredividingN.Thenn·L(A,1)/ΩAisaninteger,uptoaunitinZ[1/(2m)].