资源描述：
《A GENERALIZATION OF THE SATO–TATE CONJECTURE.pdf》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、February12,200911:59WSPC/203-IJNT00202InternationalJournalofNumberTheoryVol.5,No.1(2009)173184cWorldScientificPublishingCompanyAGENERALIZATIONOFTHESATOTATECONJECTUREWENTANGKUODepartmentofPureMathematics,FacultyofMathematicsUniversityofWaterloo,Waterloo,Ontario,CanadaN2L3G1wtkuo@math.uwaterloo.c
2、aReceived1December2006Accepted10October2007TheoriginalSatoTateConjectureconcernstheangledistributionoftheeigenvaluesarisenfromnon-CMellipticcurves.Inthispaper,weformulateananalogueoftheSatoTateConjectureonautomorphicformsofGLnand,underaholomorphicassumption,provethatthedistributioniseitherunifo
3、rmorthegeneralizedSatoTatemeasure.Keywords:L-functions;ellipticcurves;SatoTate.MathematicsSubjectClassification2000:11F03,11F251.IntroductionLetEbeanellipticcurveoverQand∆EthediscriminantofE.Forarationalprimep,p∆E,defineNp=p+1−ap=
4、E(Fp)
5、,whereE(Fp)isthesetofrationalpointsofEdefinedoverthefinitefiel
6、dFpand
7、E(Fp)
8、isthecardinalityofE(Fp).ByaresultofHasse(see[16,Theorem1.1]),wehave
9、a
10、≤2p1/2.pThuswecanwritea=2p1/2cosθ,ppwheretheangleθpisdefineduniquelywith0≤θp<π.Onecanaskhowθpdistributeintheinterval[0,π).IfEiswithcomplexmultiplication,theanswertothisquestioniswellknown(see[12]).Ontheotherhand,i
11、fEiswithoutcomplexmultiplication,theproblemremainsopenuntiltoday.Forα,β∈R,0≤α<β<π,SatoandTate(independently)conjecturedthatβ112lim·#{p
12、p≤x,α≤θp≤β}=2sinθdθx→∞π(x)πα173February12,200911:59WSPC/203-IJNT00202174W.Kuowhereπ(x)isthenumberofprimesp≤x.Itisso-calledtheSato–TateConjectureandithasmanycla
13、ssicalorigins.Forinstance,itisrelatedtohowoftenaquadraticformisaprimeinacertainregion(see[5])andhowprimesdistributeinquadraticprogressions(see[11]).RecentlyTaylorannouncedaproofoftheSato–TateCon-jectureforellipticcurveswithmultiplicativereductionsomeprime(see[15]).Thismarksagreatadvancetothepro
14、blemsincebeforehisresult,thereisnosingleellipticcurveknowntosatisfytheSato–TateConjecture.Thegoalofthispaperistostudythisconjectureinamoregeneralsetting.BytheTaniyama–Shimura–WeilConjecture,weknowthattheL-functionsofellipticcurves
