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1、ELLIPTICCURVESANDMODULARFORMSWEETECKGANThesearethenotesofagraduatecoursegiveninWinter2005atUCSD.1.AffineAlgebraicGeometryWefirstdiscusssomebasicaffinealgebraicgeometryoveranalgebraicallyclosedfieldk.Later,weshallneedtodescendtoanon-algebraicallyclosedfield.Definition:LetAnbethesetkn.Itisc
2、alledtheaffinen-spaceoverk.ItistheanalogofRnindifferentialgeometryandtheanalogofCnincomplexgeometry.WhatsortofstructurearewegoingtoendowAnwith?Forexample,isAnmerelyaset?WhattypeoffunctionsarewegoingtoconsideronAn?Inalgebraicgeometry,oneworksonlywithpolynomialfunctions,orratherwithrat
3、iosofpolynomialfunctions(theso-calledrationalfunctions).Letk[x1,...,xn]denotethepolynomialringoverkwithnvariables.Clearly,wecanthinkofelementsf∈k[x1,...,xn]asfunctionsAn−→k.OnecandefineatopologyonAn.LetIbeanidealink[x1,...,xn],andletV(I)={x∈An:f(x)=0forallf∈I}.Becausek[x1,...,xn]is
4、noetherian(Hilbert’sbasistheorem),Iisfinitelygenerated,andthusV(I)isthezerosetofafinitenumberofpolynomials.Definition:WegiveAnatopologybydecreeingthatthesetsoftheformV(I)aretheclosedsubsets.ThistopologyiscalledtheZariskitopologyonAn.Examples:InA1,theclosedsubsetsarepreciselytheemptys
5、et,thewholesetA1andfinitesetsofpoints.ObservethatA1iscompact!Exercise:Justifytheabovedefinition,i.e.showthatthecollectionofsubsetsoftheformV(I)containstheemptysetandAn,isclosedunderfiniteunionandisclosedunderarbitraryintersection.Exercise:Showthatanon-emptyopensubsetofAnisdense.Thust
6、heZariskitopologyishighlynon-Hausdorff.IstheZariskitopologyonA2thesameastheproducttopologyofA1×A1?Exercise:Clearly,theelementsofk[x1,...,xn],whenregardedasfunctionsonAn,arecontin-uouswithrespecttotheZariskitopology.ArethecontinuousfunctionsAn−→A1preciselytheelementsofk[x1,...,xn]?1
7、2WEETECKGANItwillbenecessarilytolocalizethenotionoffunctions,soweneedtoconsiderfunctionsdefinedonopensubsetsofAn.Sincek[x1,...,xn]isanintegraldomain,ithasafieldoffractionsk(x1,...,xn).Anelementofthefieldoffractionscanbeexpressedasaquotientf(x1,...,xn)/g(x1,...,xn),whichcanberegardeda
8、safunctiononthesubsetofAnwhereg6=
