关于BLUCH-TMAGAWA空间和SelMr群的一个注记

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1、ANOTEONTHEBLOCH-TAMAGAWASPACEANDSELMERGROUPSNIRANJANRAMACHANDRANABSTRACT.ForanyabelianvarietyAoveranumberfield,weconstructanextensionoftheTate-ShafarevichgroupbytheBloch-TamagawaspaceusingtherecentworkofLichtenbaumandFlach.ThisgivesanewexampleofaZagiersequenceforth

2、eSelmergroupofA.Introduction.LetAbeanabelianvarietyoveranumberfieldFandA_itsdual.B.BirchandP.Swinnerton-Dyer,interestedindefiningtheTamagawanumber(A)ofA,wereledtotheircelebratedconjecture[2,Conjecture0.2]fortheL-functionL(A;s)(ofAoverF)whichpredictsbothitsorderrofv

3、anishinganditsleadingtermcAats=1.Thedifficultyindefining(A)directlyisthattheadelicquotientA(AF)isHausdorffonlywhenr=0,i.e.,A(F)isfinite.S.BlochA(F)[2]hasintroducedasemiabelianvarietyGoverFwithquotientAsuchthatG(F)isdiscreteandcocompactinG(AF)[2,Theorem1.10]andfamous

4、lyproved[2,Theorem1.17]thattheTamagawanumberconjecture-recalledbrieflybelow,see(5)-forGisequivalenttotheBirch-Swinnerton-DyerconjectureforAoverF.ObservethatGisnotalinearalgebraicgroup.TheBloch-TamagawaspaceX=G(AF)ofA=FiscompactandHausdorff.AG(F)Theaimofthisshortnot

5、eistoindicateafunctorialconstructionofalocallycompactgroupYA(1)0!XA!YA!Ш(A=F)!0;anextensionoftheTate-ShafarevichgroupШ(A=F)byXA.ThecompactnessofYAisclearlyequivalenttothefinitenessofШ(A=F).ThisconstructionwouldbestraightforwardifG(L)werediscreteinG(AL)forallfiniteex

6、tensionsLofF.Butthisisnottrue(Lemma4):thequotientG(AL)G(L)isnotHausdorff,ingeneral.TheverysimpleideafortheconstructionofYAis:Yoneda’slemma.Namely,weconsiderthecategoryoftopologicalG-modulesasasubcategoryoftheclassifyingtoposBGofG(naturalfromthecontextofthecontinuo

7、uscohomologyofatopologicalgroupG,asinS.Lichtenbaum[10],M.Flach[5])andconstructYAviatheclassifyingtoposoftheGaloisgroupofF.D.Zagier[18]haspointedoutthattheSelmergroupsSelm(A=F)(6)canbeobtainedfromcertaintwo-extensions(7)ofШ(A=F)byA(F);thesewecallZagiersequences.Wes

8、howhowYAprovidesanewnaturalZagiersequence.Inparticular,thisshowsthatYAisnotasplitsequence.TheZagiersequenceA(AF)0!A(F)!A(AF)!()!Ш(A=F)!0A(F)seemstobe