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1、I.AutomorphicrepresentationsandGaloisrepresentationsIwillbelecturingonjointworkinprogresswithRichardTaylor,andinpartwithLaurentClozel.Ourgoalistoprovethatcertainkindsoffinite-dimensionalrepresentationsρofGE=Gal(E/E¯),whereEisanumberfield,areassociatedtoautomorphicforms.Mygo
2、alinthefirsttalkoftheseriesistomakethegoalofourworkinprogresscomprehensible.Thismeans,inparticular,addressingthefollowingquestions:1.WhatkindsofrepresentationsofGEdowehaveinmind?2.Whatpropertiesdotherepresentationshavetohaveinordertobeassoci-atedtoautomorphicforms,andforwh
3、atEisthisconceivable?3.Whatkindsofautomorphicformshaverepresentationsassociatedtothem,andhow?4.Whatdoes“associated”mean,anyway?Afifthquestion,namelywhyonewouldwanttolookatrepresentationsofGEatall,isaddressedinmyJuniorColloquiumtalk.Herearetheshortanswers:1.Ans.ρisan`-adicr
4、epresentation,i.e.ann-dimensionalrepresentationonavectorspaceoverafiniteextensionofQ`,where`issomeprimenumber.2.Ans.Forthetimebeing,EhastobetotallyrealoraCMfield,andρhastobeofgeometrictypeintheterminologyofFontaine-Mazur.Iwilltrytoexplainthisslowly.ThisisthefullscopeoftheLa
5、nglandsconjectures.Fortheforeseeablefuture,onehastomakeadditionalrestrictiveassumptionsontheGaloisrepresentations.3.Ans.ThemodelisalwaysthetheoremofEichler-Shimura,whichattachestwo-dimensional`-adicrepresentationstoholomorphicmodularformsofweight2,anditsgeneralizationbyDe
6、lignetoweightsk≥2.FormoregeneralGaloisrepresentationsonehastoworkwithautomorphicrepre-sentationsπofGL(n,E)–Iwillsaysomethingaboutwhatthismeans–ratherthanindividualautomorphicforms.TheclassofπwhichonecanhopetoassociatetoGaloisrepresentationswasidentifiedinClozel’sAnnArborar
7、ticle;theyarethoseofalgebraictype.Paralleltotheadditionalre-strictionsontheρareadditionalrestrictionsontheπ;thesearethenaturalgeneralizationsofmodularformsofweightk≥2,butrecognizingthemas“natural”takessomepractice.4.Ans.Bothρandtheautomorphicrepresentationπρcanbeidentified
8、fromtheirL-functionsL(s,ρ),L(s,πρ)whichareDirichletserieswithEulerprod-ucts.Theyareassociatedift
