a (very brief) History of the Trace Formula-James Arthur.pdf

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1、A(verybrief)HistoryoftheTraceFormulaJamesArthurThisnoteisashortsummaryofalectureintheseriescelebratingthetenthanniversaryofPIMS.Thelectureitselfwasanattempttointroducethetraceformulathroughitshistoricalorigins.IthankBillCasselmanforsuggestingthetopic.IwouldalsoliketothankPeterSarna

2、kforsharinghishistoricalinsightswithme.IhopeIhavenotdistortedthemtoogrievously.Asitispresentlyunderstood,thetraceformulaisageneralidentityXX(GTF)fgeometrictermsg=fspectraltermsg:Thespectraltermscontainarithmeticinformationofafundamentalnature.However,theyarehighlyinaccessible,spec

3、tral"actually,inthenonmathematicalmeaningoftheword.Thegeometrictermsarequiteexplicit,buttheyhavethedrawbackofbeingverycomplicated.Therearesimpleanaloguesofthetraceformula,toymodels"onecouldsay,whicharefamiliartoall.Forexample,supposethatA=(aij)isacomplex(nn)-matrix,withdiagonalen

4、triesfuig=faiigandeigenvaluesfjg.Byevaluatingitstraceintwodierentways,weobtainanidentityXnXnui=j:i=1j=1ThediagonalcoecientsobviouslycarrygeometricinformationaboutAasatransformationofCn.Theeigenvaluesarespectral,intheprecisemathematicalsenseoftheword.Foranotherexample,supposetha

5、tg2C1(Rn).ThisfunctioncthensatisesthePoissonsummationformulaXXg(u)=g^();u2Zn22iZnwhereZg^()=g(x)exdx;2Cn;RnistheFouriertransformofg.Oneobtainsaninterestingapplicationbylettingg=gTapproximatethecharacteristicfunctionoftheclosedballBTofradiusTabouttheorigin.AsTbecomeslarge,th

6、elefthandsideapproximatesthenumberoflatticepointsu2ZninB.TThedominanttermontherighthandsideistheintegral12Zg^(0)=g(x)dx;Rnwhichinturnapproximatesvol(BT).Inthisway,thePoissonsumma-tionformulaleadstoasharpasymptoticformulaforthenumberoflatticepointsinBT.Ourrealstartingpointistheupper

7、halfplaneH=z2C:Im(z)>0:ThemultiplicativegroupSL(2;R)of(22)realmatricesofdetermi-nant1actstransitivelybylinearfractionaltransformationsonH.Thediscretesubgroup=SL(2;Z)actsdiscontinuously.ItsspaceoforbitsnHcanbeidentiedwithanoncompactRiemannsurface,whosefundamentaldomainisthefami

8、l-iarmodularregion.1-101Mo