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1、UNEARTHINGTHEVISIONSOFAMASTER:HARMONICMAASSFORMSANDNUMBERTHEORYKENONOFormyteacherBasilGordononhis75thbirthdayAbstract.Togetherwithhiscollaborators,mostnotablyKathrinBringmannandJanBruinier,theauthorhasbeenresearchingharmonicMaassforms.Thesenon-holomorph
2、icmodularformsplaycentralrolesinmanysubjects:arithmeticgeometry,combinatorics,modularforms,andmathematicalphysics.Hereweoutlinethegeneralfacetsofthethe-ory,andwegiveseveralapplicationstonumbertheory:partitionsandq-series,modularforms,singularmoduli,Borc
3、herdsproducts,extensionsoftheoremsofKohnen-ZagierandWaldspurgeronmodularL-functions,andtheworkofBruinierandYangonGross-Zagierformulae.Whatissurprisingisthatthisstoryhasanunlikelybeginning:thepursuitofthesolutiontoagreatmathematicalmystery.Modularformsar
4、ecentralincontemporarymathematics.Indeed,modularformsplaycrucialrolesinalgebraicnumbertheory,algebraictopology,arithmeticgeometry,com-binatorics,numbertheory,representationtheory,andmathematicalphysics.Therecenthistoryofthesubjectincludes(tonameafew)gre
5、atsuccessesontheBirchandSwinnerton-DyerConjecture,MirrorSymmetry,MonstrousMoonshine,andtheproofofFermat’sLastTheorem.Thesecelebratedworksaredramaticexamplesoftheevolutionofmathematics;indeed,itwouldhavebeenimpossibletoprophesythemfiftyyearsago.Insteadoft
6、ravellingbackintimetothe1950s,ourstory(alsosee[165])beginsin1887,1inavillageinIndia.Ourmathematics,whichisaboutharmonicMaassforms,beginswiththelegend(see[6,39,40,110,111,133,165])ofthegreatmathematicianSrinivasaRamanujan,andthemathematicsheconjuredfromh
7、isdeathbed.1.ThegiftfromKumbakonamRamanujanwasbornonDecember22,1887inErode,atownabout250milessouthwestofChennai(formerlyknownasMadras).HewasaBrahmin,amemberofIndia’spriestlycaste.HegrewupinKumbakonam,asmalltownroughly150milessouthofChennai,wherehisfathe
8、rSrinivasawasaclothmerchant’sclerk.Kumbakonam,whichissituatedonthebanksofthesacredCauveryRiver,was(andremainstoday)acosmopolitancenteroftheruralIndiandistrictofTanjoreinthestateofTamilNadu.2Asaboy,Ramanujanwasastellarstudent.Inde
