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Non-abelian Fundamental Groups and Iwasawa Theory PDF

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LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL, UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat http://www.cambridge.org/mathematics 287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MARTÍNEZ(eds) 288 Surveysincombinatorics,2001,J.W.P.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE 290 QuantumgroupsandLietheory,A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed) 292 Aquantumgroupsprimer,S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK 294 Introductiontooperatorspacetheory,G.PISIER 295 Geometryandintegrability,L.MASON&Y.NUTKU(eds) 296 Lecturesoninvarianttheory,I.DOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES 298 Higheroperads,highercategories,T.LEINSTER(ed) 299 Kleiniangroupsandhyperbolic3-manifolds,Y.KOMORI,V.MARKOVIC&C.SERIES(eds) 300 IntroductiontoMöbiusdifferentialgeometry,U.HERTRICH-JEROMIN 301 StablemodulesandtheD(2)-problem,F.E.A.JOHNSON 302 DiscreteandcontinuousnonlinearSchrödingersystems,M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH 303 Numbertheoryandalgebraicgeometry,M.REID&A.SKOROBOGATOV(eds) 304 GroupsStAndrews2001inOxfordI,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 305 GroupsStAndrews2001inOxfordII,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 306 Geometricmechanicsandsymmetry,J.MONTALDI&T.RATIU(eds) 307 Surveysincombinatorics2003,C.D.WENSLEY(ed.) 308 Topology,geometryandquantumfieldtheory,U.L.TILLMANN(ed) 309 Coringsandcomodules,T.BRZEZINSKI&R.WISBAUER 310 Topicsindynamicsandergodictheory,S.BEZUGLYI&S.KOLYADA(eds) 311 Groups:topological,combinatorialandarithmeticaspects,T.W.MÜLLER(ed) 312 Foundationsofcomputationalmathematics,Minneapolis2002,F.CUCKERetal(eds) 313 Transcendentalaspectsofalgebraiccycles,S.MÜLLER-STACH&C.PETERS(eds) 314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC´,P.ROWLINSON&S.SIMIC´ 315 Structuredringspectra,A.BAKER&B.RICHTER(eds) 316 Linearlogicincomputerscience,T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds) 317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds) 318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations,D.HENRY 319 DoubleaffineHeckealgebras,I.CHEREDNIK 320 L-functionsandGaloisrepresentations,D.BURNS,K.BUZZARD&J.NEKOVÁRˇ(eds) 321 Surveysinmodernmathematics,V.PRASOLOV&Y.ILYASHENKO(eds) 322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds) 323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds) 324 Singularitiesandcomputeralgebra,C.LOSSEN&G.PFISTER(eds) 325 LecturesontheRicciflow,P.TOPPING 326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS 327 Surveysincombinatorics2005,B.S.WEBB(ed) 328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds) 329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds) 330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed) 331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SÜLI&M.J.TODD (eds) 332 Handbookoftiltingtheory,L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds) 333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK 334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN 335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER 336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE 337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds) 338 Surveysingeometryandnumbertheory,N.YOUNG(ed) 339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C. SNAITH(eds) 342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds) 343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds) 344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds) 345 Algebraicandanalyticgeometry,A.NEEMAN 346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds) 347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds) 348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds) 349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY& A.WILKIE(eds) 350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY& A.WILKIE(eds) 351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH 352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds) 353 Trendsinstochasticanalysis,J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds) 354 Groupsandanalysis,K.TENT(ed) 355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI 356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO 357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds) 358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER&I.J.LEARY (eds) 359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA& S.RAMANAN(eds) 360 Zariskigeometries,B.ZILBER 361 Words:Notesonverbalwidthingroups,D.SEGAL 362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERÓN&R.ZUAZUA 363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS& M.J.TODD(eds) 364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds) 365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds) 366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds) 367 Randommatrices:Highdimensionalphenomena,G.BLOWER 368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds) 369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIÉ 370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH 371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI 372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds) 373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC 375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds) 377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY 378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds) 379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds) 380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds) 381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&P.WINTERNITZ (eds) 382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCˇEK 383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS, J.NICAISE&J.SEBAG(eds) 384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS, J.NICAISE&J.SEBAG(eds) 385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN& T.WEISSMAN(eds) 386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER 387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal(eds) 388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal(eds) 389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI 390 Localizationinperiodicpotentials,D.E.PELINOVSKY 391 FusionsystemsinalgebraandtopologyM.ASCHBACHER,R.KESSAR&B.OLIVER 392 Surveysincombinatorics2011,R.CHAPMAN(ed) 393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATES,M.KIM,F.POP,M.SAIDI&P.SCHNEIDER (eds) 394 Variationalproblemsindifferentialgeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds) LondonMathematicalSocietyLectureNoteSeries:393 Non-abelian Fundamental Groups and Iwasawa Theory Editedby JOHN COATES UniversityofCambridge MINHYONG KIM UniversityCollegeLondon FLORIAN POP UniversityofPennsylvania MOHAMED SAIDI UniversityofExeter PETER SCHNEIDER UniversitätMünster CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9781107648852 (cid:2)c CambridgeUniversityPress2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2012 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata Non-abelianfundamentalgroupsandIwasawa theory/editedbyJohnCoates...[etal.]. p. cm.–(LondonMathematicalSocietylecturenoteseries;393) ISBN978-1-107-64885-2(pbk.) 1. Iwasawatheory. 2. Non-Abeliangroups. I. Coates,J. II. Title. III. Series. QA247.N56 2011 (cid:3) 512.74–dc23 2011027955 ISBN978-1-107-64885-2Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Listofcontributors pagevi Preface vii Lecturesonanabelianphenomenaingeometryandarithmetic FlorianPop 1 OnGaloisrigidityoffundamentalgroupsofalgebraiccurves HiroakiNakamura 56 AroundtheGrothendieckanabeliansectionconjecture MohamedSa¨ıdi 72 FromtheclassicaltothenoncommutativeIwasawatheory (fortotallyrealnumberfields) MaheshKakde 107 OntheM (G)-conjecture H J.CoatesandR.Sujatha 132 GaloistheoryandDiophantinegeometry MinhyongKim 162 Potentialmodularity–asurvey KevinBuzzard 188 RemarksonsomelocallyQ -analyticrepresentationsofGL (F) p 2 inthecrystallinecase ChristopheBreuil 212 Completedcohomology–asurvey FrankCalegariandMatthewEmerton 239 Tensorandhomotopycriteriaforfunctionalequationsof(cid:2)-adic andclassicaliteratedintegrals HiroakiNakamuraandZdzisławWojtkowiak 258 v Contributors ChristopheBreuil Baˆtiment425,C.N.R.S.etUniversite´ Paris-Sud, 91405OrsayCedex,France KevinBuzzard DepartmentofMathematics,ImperialCollegeLondon, 180Queen’sGate,LondonSW72AZ,UK FrankCalegari DepartmentofMathematics,NorthwesternUniversity, 2033SheridanRoad,Evanston,IL60208-2730,USA J. Coates DPMMS, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB30WB,UK Matthew Emerton Department of Mathematics, Northwestern University, 2033SheridanRoad,Evanston,IL60208-2730,USA Mahesh Kakde Department of Mathematics, University College London, GowerStreet,LondonWC1E6BT,UK Minhyong Kim Department of Mathematics, University College London, GowerStreet,LondonWC1E6BT,UK HiroakiNakamura DepartmentofMathematics, OkayamaUniversity,Okayama700-8530,Japan FlorianPop DepartmentofMathematics,UniversityofPennsylvania, 209South33rdStreet,Philadelphia,PA19104-6395,USA MohamedSa¨ıdi CollegeofEngineering,Mathematics,andPhysical Sciences,UniversityofExeter,HarrisonBuilding,NorthParkRoad, ExeterEX44QF,UK R.Sujatha SchoolofMathematics,TataInstituteofFundamentalResearch, HomiBhabhaRoad,Mumbai400005,India ZdzisławWojtkowiak Universite´ deNice-SophiaAntipolis,De´pt.ofMath., LaboratoireJeanAlexandreDieudonne´,U.R.A.auC.N.R.S.,No168, ParcValrose–B.P.N◦71,06108NiceCedex2,France vi Preface Inhistoricalorigins,serioustopologicalinputtoarithmeticstartsatthelatest in the early years of the twentieth century with the foundations of class field theorybyTakagiandArtin.Aroundthemiddleofthecentury,therapiddevel- opmentofhomologicaltechniquesinalgebraledtoanexplosionofconsequent activityinnumbertheoryandalgebraicgeometry,includingthetheoryofco- herentsheaves,thehomologicalinterpretationofArtin’sreciprocitymap,and Grothendieck’sconstructionofarithmeticcohomologytheoriesbaseduponthe abstractnotionofGrothendiecktopologies.Then,inthe1980s,Grothendieck formulated his anabelian conjectures in the celebrated letter to Faltings, and brought to a hitherto-unexplored depth the interaction between topology and arithmetic. Therein came into focus the far-reaching vision that the perspec- tive of non-abelian fundamental groups could lead to a fundamentally new understandingofdeeparithmeticphenomena,includingthearithmetictheory ofmoduliandDiophantinefinitenessonhyperboliccurves.Inthe1990s,pro- found (and perhaps unexpected) progress in Grothendieck’s program was re- alized through the theorems of Nakamura, Tamagawa, Mochizuki. (See the contributionsofNakamura,Pop,andSaidi.) Meanwhile, a certain amount of work in recent years linking fundamental groups to Diophantine geometry intimates deep and mysterious connections to the theory of motives and Iwasawa theory, notions usually associated to Diophantineproblemsinvolvingexactformulaeofwhichthemostcelebrated example is the conjecture of Birch and Swinnerton-Dyer (cf. Kim’s article). Infact,theworkthusfarsuggeststhatthestill-unresolvedsectionconjecture of Grothendieck, whereby maps from Galois groups of number fields to fun- damentalgroupsofhyperbolicarithmeticcurvesareallproposedtobeofge- ometric origin, is exactly the sort of key problem that touches the core of all these areas of number theory and more. Therefore, the time seems right to vii viii Preface encourageamuchbroaderunderstandingofthearithmeticissuessurrounding anabeliangeometryanditsramifications. Whiletheoverallimportanceofthetheoremsofanabeliangeometryappears tobewidelyacknowledged,thereisasyetnotmuchspecificknowledgewithin thearithmeticgeometrycommunityofitscoherentbodyofconceptsandphi- losophy,andofthenewtechnologythatyieldsactualresults.Itisourbeliefthat ahigherlevelofgeneralawarenesswillleadtoagenuinestrengtheningofour grasp of a wide range of Galois-theoretic and Diophantine phenomena, and hopefully to significant progress on the section conjecture. The goal then of thisbookistopresentarticlesthatcontaintheideasandproblemsofanabelian geometry within the global context of mainstream arithmetic geometry, with strongemphasisonconnectionstonon-commutativeIwasawatheory. Broadlyput,anabeliangeometryandnon-commutativeIwasawatheorydeal indifferentwayswiththeliftingofhomologicalandabelianideasofarithmetic into the realm of the non-abelian, with connections to homotopy theory. The complementarynatureofthetwoviewpointsisexpressedinpartbytheexact sequence (*) 0→πˆ (X¯,b)→πˆ (X,b)→Gal(Q¯/Q)→0 1 1 associatedtoanalgebraiccurveXdefinedovertherationals,involvingthegeo- metricfundamentalgroupπˆ (X¯,b)andtheabsoluteGaloisgroupGal(Q¯/Q)of 1 therationals,bothcontributinginintricatelyinter-relatedwaystothestructure ofthearithmeticfundamentalgroupπˆ (X,b)inthemiddle.Fromthisexactse- 1 quence,oneextractsaninducedouteractionofGal(Q¯/Q)onπˆ (X¯,b).When 1 this actionis viewed atthe level ofthe abelianization H (X¯,Zˆ) of πˆ (X¯,b), it 1 1 extendsnaturallytocompletedgroupalgebrasofquotientgroupsofGal(Q¯/Q), whichthenbecomethecentralobjectsofstudyinnon-commutativeIwasawa theory(cf.articlesofCoatesandSujatha,andofKakde). Here,encodingtheinfinitetowerofnumberfieldsrepresentedbyaninfinite non-abelianquotientintoasinglegroupalgebraisbothconvenientandcrucial, and brings with it the intervention of refined tools that combine the structure theoryofnon-commutativealgebraswithcohomologicalaspectsofGaloisthe- ory.Inanabeliangeometry,ontheotherhand,themainpointistostudythefirst group πˆ (X¯,b) in its own right, without abelianizing, even as certain abelian 1 or mildlynon-abelian quotientsor completionsmight mediatethis study.Al- thoughtheattendantactionofGal(Q¯/Q)isofkeyimportance,itissafetosay thattheoverwhelminglygeometricnatureofthetechniquesaswellasthenon- linearity of the action tend to obscure the role of fine invariants of Gal(Q¯/Q) suchasmightoccurinIwasawatheory.Thereby,withsomeoversimplification, one could describe both anabelian geometry and non-commutative Iwasawa Preface ix theory within the single framework of the sequence (*), the difference lying onlyinthetechnicaldepthswithwhichattentionisdrawntoeitherthekernel or the quotient, and in the degree of non-commutativity/non-linearity that is preservedinitsstudy. Havingformulatedthusthecommonalityanddifference,theimportanceof coming up with an approach unified at both the philosophical and the tech- nologicallevelbecomesquiteobvious.Asdiscussedinanumberofdifferent chaptersofthisbook,thereisaremarkableoverlapbetweentheanabelianand Iwasawa-theoreticviewpointsinthestudyofellipticcurves.Ontheotherhand, when the geometric fundamental group is genuinely non-abelian, as happens forhyperboliccurves,itremainstodevelopthecorrectanalogueofthisinter- action.Inadditiontoprogressonthesectionconjectureitself,itishopedthat an extended program of the Birch and Swinnerton-Dyer type for hyperbolic curves will result from investigation along these lines. Because these issues lieatthecrossroadsofimportantresearchinperhapsallofthemainareasof arithmeticgeometry,ourintentionisthatthecorescientificinteractionofthe book should be enriched further by surrounding essays (by Breuil, Buzzard, andCalegariandEmerton)thatdealwithoneotherkeyarenafornon-abelian ideasandtechniques,namely,theLanglandsprogram. The articles in this book were written by participants in the special pro- grammeon‘non-abelianfundamentalgroupsinarithmeticgeometry’thattook placeattheIsaacNewtonInstituteinthesecondhalfoftheyear2009.Inthe course of the programme, we benefitted greatly from the advice of director David Wallace as well as the hospitality of the INI staff, especially Christine West,EsperanzadeFelipe,andMustaphaAmrani.Wearedeeplyindebtedto theirefficienthelp.ManythanksareduealsotoDianaGilloolyofCambridge University Press, whose encouragement was critical to the inception of this book and whose sterling effort to bring together the disparate contributions intoasinglemanuscriptwasessentialtoitscompletion.

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Number theory currently has at least three different perspectives on non-abelian phenomena: the Langlands programme, non-commutative Iwasawa theory and anabelian geometry. In the second half of 2009, experts from each of these three areas gathered at the Isaac Newton Institute in Cambridge to explai
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