资源描述：
《ARITHMETIC CHERN-SIMONS THEORY I》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、ARITHMETICCHERN-SIMONSTHEORYIMINHYONGKIMwithAppendixBbyBehrangNoohiAbstract.Inthispaper,weapplyideasofDijkgraafandWitten[24,6]on2+1dimensionaltopologicalquantumfieldtheorytoarithmeticcurves,thatis,thespectraofringsofintegersinalgebraicnumberfields.Inthefirstthreese
2、ctions,wedefineclassicalChern-SimonsfunctionalsonspacesofGaloisrepresentations.Inthehighlyspeculativesection6,weconsiderthefar-fetchedpossibilityofusingChern-SimonstheorytoconstructL-functions.1.ThearithmeticChern-Simonsaction:basiccaseWewishtomoveratherquicklyto
3、aconcretedefinitioninthisfirstsection.Thereaderisdirectedtosection5foramotivationaldiscussionofL-functions.LetX=Spec(OF),thespectrumoftheringofintegersinanumberfieldF.WeassumethatFistotallyimaginary,forsimplicityofexposition.DenotebyGmthe´etalesheafthatassociatesto
4、aschemetheunitsintheglobalsectionsofitscoordinatering.Wehavethefollowingcanonicalisomorphism([19],p.538):inv:H3(X,G)≃Q/Z.(∗)mThismapisdeducedfromthe‘invariant’mapoflocalclassfieldtheory.Wewillusethesamenameforarangeofisomorphismshavingthesameessentialnature,forex
5、ample,3inv:H(X,Zp(1))≃Zp,(∗∗)whereZp(1)=lim←−µpi,andµn⊂Gmisthesheafofn-throotsof1.Thisfollowsifromtheexactsequencen(·)n0→µn→Gm→Gm→Gm/(Gm)→0.arXiv:1510.05818v4[math.NT]11Nov2016Thatis,accordingtoloc.cit.,H2(X,G)=0,mwhilebyop.cit.,p.551,wehaveHi(X,G/(G)n)=0mmfori≥
6、1.Ifwebreakuptheaboveintotwoshortexactsequences,n(·)0→µn→Gm→Kn→0,and0→K→G→G/(G)n→0,nmmm1991MathematicsSubjectClassification.14G10,11G40,81T45.M.K.SupportedbygrantEP/M024830/1fromtheEPSRC.12MINHYONGKIMwededuceH2(X,K)=0,nfromwhichitfollowsthat31H(X,µn)≃Z/Z,nthen-to
7、rsioninsideQ/Z.Takingtheinverselimitovern=pigivesthesecondisomorphismabove.Thepro-sheafZp(1)isaveryfamiliarcoefficientsystemfor´etalecohomologyand(**)isreminiscentofthefundamentalclassofacompactorientedthreemanifoldforsingularcohomology.SuchananalogywasnotedbyMazu
8、raround50yearsago[20]andhasbeendevelopedrathersystematicallybyanumberofmathematicians,notably,MasanoriMorishita[21].Withinthiscircleofideasisincludedtheanalogybetween
