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SERRE'S MODULARITY CONJECTURE I

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SERRE'S MODULARITY CONJECTURE (I) CHANDRASHEKHAR KHARE AND JEAN-PIERRE WINTENBERGER to Jean-Pierre Serre Abstract. This paper is the first part of a work which proves Serre's modularity conjecture. We first prove the cases p 6= 2 and odd con- ductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [18]. We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin [24]. Contents 1. Introduction 2 1.1. Main result 2 1.2. The nature of the proof of Theorem 1.2 3 1.3. A comparison to the approach of [17] 3 1.4. Description of the paper 4 1.5. Notation 4 2. A crucial definition 4 3. Proof of Theorem 1.2 5 3.1. Auxiliary theorems 5 3.2. Proof of Theorem 1.2 6 4. Modularity lifting results 6 5. Compatible systems of geometric representations 7 6. Some utilitarian lemmas 10 7. Estimates on primes 12 8. Proofs of the auxiliary theorems 12 8.1. Proof of Theorem 3.1 12 8.2. Proof of Theorem 3.2 12 8.3. Proof of Theorem 3.3 15 8.4. Proof of Theorem 3.4 16 9. The general case 18 10. Modularity of compatible systems 19 11. Acknowledgements 20 References 20 CK was partially supported by NSF grants DMS 0355528 and DMS 0653821, the Miller Institute for Basic Research in Science, University of California Berkeley, and a Guggenheim fellowship.

  • reducible continuous representation

  • lifting theorems

  • continuous irreducible representation

  • theorems stated

  • vv

  • main theorem

  • theorems

  • modularity

  • prime divisors


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SERRE’SMODULARITYCONJECTURE(I)CHANDRASHEKHARKHAREANDJEAN-PIERREWINTENBERGERtoJean-PierreSerreAbstract.ThispaperisthefirstpartofaworkwhichprovesSerre’smodularityconjecture.Wefirstprovethecasesp6=2andoddcon-ductor,seeTheorem1.2,moduloTheorems4.1and5.1.Theorems4.1and5.1areproveninthesecondpart,see[18].Wethenreducethegeneralcasetoamodularitystatementfor2-adicliftsofmodularmod2representations.ThisstatementisnowatheoremofKisin[24].Contents1.Introduction21.1.Mainresult21.2.ThenatureoftheproofofTheorem1.231.3.Acomparisontotheapproachof[17]31.4.Descriptionofthepaper41.5.Notation42.Acrucialdenition43.ProofofTheorem1.253.1.Auxiliarytheorems53.2.ProofofTheorem1.264.Modularityliftingresults65.Compatiblesystemsofgeometricrepresentations76.Someutilitarianlemmas107.Estimatesonprimes128.Proofsoftheauxiliarytheorems128.1.ProofofTheorem3.1128.2.ProofofTheorem3.2128.3.ProofofTheorem3.3158.4.ProofofTheorem3.4169.Thegeneralcase1810.Modularityofcompatiblesystems1911.Acknowledgements20References20CKwaspartiallysupportedbyNSFgrantsDMS0355528andDMS0653821,theMillerInstituteforBasicResearchinScience,UniversityofCaliforniaBerkeley,andaGuggenheimfellowship.JPWismemberoftheInstitutUniversitairedeFrance.1
2CHANDRASHEKHARKHAREANDJ-P.WINTENBERGER1.IntroductionLetGQ=Gal(Q¯/Q)betheabsoluteGaloisgroupofQ.Letρ¯:GQGL2(F)beacontinuous,absolutelyirreducible,two-dimensional,odd(detρ¯(c)=1forcacomplexconjugation),modprepresentation,withFafinitefieldofcharacteristicp.WesaythatsucharepresentationisofSerre-type,orS-type,forshort.WedenotebyN(ρ¯)the(primetop)Artinconductorofρ¯,andk(ρ¯)theweightofρ¯asdefinedin[33].Itisanimportantfeatureoftheweightk(ρ¯),forp>2,thatifχpisthemodpcyclotomiccharacter,thenforsomeiZ,2k(ρ¯χpi)p+1.Inthecaseofp=2,thevaluesofk(ρ¯)caneitherbe2or4,withtheformerifandonlyifρ¯isfiniteat2.Wefixembeddingsιp:Q֒Qpforallprimesphereafter,andwhenwesay(aplaceabove)p,wewillmeantheplaceinducedbythisembedding.Serrehasconjecturedin[33]thatsuchaρ¯ismodularofweightk(ρ¯)andlevelN(ρ¯),i.e.,arisesfrom(withrespecttothefixedembeddingιp:Q֒Qp)anewformfofweightk(ρ¯)andlevelN(ρ¯).Byarisesfromfwemeanthatthereisanintegralmodelρ:GQGL2(O)ofthep-adicrepresentationρfassociatedtof,suchthatρ¯isisomorphictothereductionofρmodulothemaximalidealofO,andwithOtheringofintegersofafiniteextensionofQp.Inthesecircumstanceswealsosaythatρ¯arisesfromSk(ρ¯)1(N(ρ¯))).1.1.Mainresult.Thecaseoftheconjectureforconductorone,i.e.,theleveloneconjecture,wasprovedin[19].Theorem1.1.Aρ¯ofS-typewithN(ρ¯)=1arisesfromSk(ρ¯)(SL2(Z)).Thisbuiltontheideasintroducedin[17].InthispaperwefirstextendTheorem1.1,andthemethodsofitsproof,andprovethefollowingtheorem.Theorem1.2.1.Letpbeanoddprime.Thenaρ¯ofS-typewithN(ρ¯)anoddintegerarisesfromSk(ρ¯)1(N(ρ¯))).2.Letp=2.Thenaρ¯ofS-typewithk(ρ¯)=2arisesfromS21(N(ρ¯))).WenotethatTheorem1.2(2)alsocompletestheworkthatthequalitativeformofSerre’sconjectureimpliestherefinedformbyfillinginamissingcaseincharacteristic2:controlofthelevelforρ¯|D2scalarwhentheprojectveimageofρ¯isnotdihedral(see[3]and[36]).Nevertheless,wedonotknowforp=2thestrongformoftheconjectureinthesenseofEdixhoven:wedonotknowingeneralthatρ¯isunramifiedat2ifandonlyifitarisesfromaKatzformofweight1([11]).WereduceinTheorem9.1thegeneralcaseofSerre’sconjecturetoacertainhypothesis(H)whichisnowatheoremofKisin,see[24].InTheorem9.1,assuming(H),wefirstprovethecasep=2,k(ρ¯)=4,andthenwededucefromitthecasep6=2andN(ρ¯)even.InthispartwewillproveTheorem1.2modulotwoliftingtheorems,The-orem4.1(closelyrelatedtoTheorem6.1of[19])andTheorem5.1(closely