欢迎来到天天文库
浏览记录
ID:39078169
大小:204.77 KB
页数:7页
时间:2019-06-24
《关于BLUCH-TMAGAWA空间和SelMr群的一个注记》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
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
此文档下载收益归作者所有