Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge A Series of Modern Surveys in Mathematics 61 Werner Lütkebohmert Rigid Geometry of Curves and Their Jacobians Ergebnisse der Mathematik Volume 61 und ihrer Grenzgebiete 3.Folge A Series of Modern Surveys in Mathematics EditorialBoard L.Ambrosio,Pisa V.Baladi,Pariscedex05 G.-M.Greuel,Kaiserslautern M.Gromov,Bures-sur-Yvette G.Huisken,Tübingen J.Jost,Leipzig J.Kollár,Princeton S.S.Kudla,Toronto G.Laumon,OrsayCedex U.Tillmann,Oxford J.Tits,Paris D.B.Zagier,Bonn Forfurthervolumes: www.springer.com/series/728 Werner Lütkebohmert Rigid Geometry of Curves and Their Jacobians WernerLütkebohmert InstituteofPureMathematics UlmUniversity Ulm,Germany ISSN0071-1136 ISSN2197-5655(electronic) ErgebnissederMathematikundihrerGrenzgebiete.3.Folge/ASeriesofModernSurveys inMathematics ISBN978-3-319-27369-3 ISBN978-3-319-27371-6(eBook) DOI10.1007/978-3-319-27371-6 LibraryofCongressControlNumber:2016931305 MathematicsSubjectClassification(2010): 14G22,14H40,14K15,30G06 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Enhommageà MichelRaynaud Preface Projectivealgebraiccurvesorabelianvarietiesaredefinedasthevanishinglocusof finite families of homogeneous polynomials in a projective space fulfilling certain conditions. Except for elliptic curves or hyperelliptic curves, it is difficult to pin downequationswhichgiverisetocurvesorabelianvarieties. Overthecomplexnumbersonehasanalytictoolstoconstructandtouniformize suchobjects.Forexample,everysmoothcurveofgenusg≥2hasarepresentation Γ\H, where H is the upper half-plane and Γ ⊂Aut(H) is a group acting on H. Similarly,everycompactcomplexLiegroupisoftypeCn/Λ,whereΛisalattice inCn;theabelianvarietiesamongthecompactcomplexLiegroupscanbecharac- terizedviapolarizations.Moreover,one canconstructcurves andabelianvarieties inthiswayviaalgebraizationoftheanalyticquotients.Thus,thegeometryandthe constructionofsuchobjectsarecompletelyclarified. Overacompletefield K withrespecttoanon-Archimedeanvaluation,onecan expect similar tools as in the complex case once a good theory of holomorphic functionshasbeenestablished. Historically,thetheorystartedwiththesimplestcaseofanellipticcurveoverK. OnecandefinetheellipticcurvebyaminimalWeierstraßequationwithintegralco- efficients.Ifthisequationreducestoanellipticcurveovertheresiduefield,wesay thatthegivenellipticcurvehasgoodreduction.Inthiscasethereisnouniformiza- tionatall;suchcurvescanberegardedasliftingsofellipticcurvesdefinedoverthe residuefield.Ontheotherhand,iftheWeierstraßequationreducestoacubicwith anordinarydoublepoint,thenthesituationlooksbetterfromtheviewpointofuni- formization.AsanabstractgroupitsK-rationalpointsarerepresentedbyaquotient K×/qZforsomenon-integralq∈K×withoutanyfurtherstructure.OriginallyTate wantedtoconstruct“analytic”quotientsG /qZ ofthemultiplicativegroupofa m,K Z non-ArchimedeanfieldK bythelatticeq ;aconstructionwhichcannotbecarried outinthecategoryofordinaryschemesdirectly. Thus, there was the desire to create a theory of “analytic spaces” over a non- Archimedean field which allows such constructions. This was exactly the incen- tiveofTatetounderstandellipticcurveswithmultiplicativereductionby“analytic” vii viii Preface means.In1961TategaveaseminaratHarvardwherehedevelopedatheoryofrigid analyticspaces;cf.[92]. Lateron,usingmethodsfromformalalgebraicgeometry,Mumfordgeneralized the construction of Tate’s elliptic curve to curves of higher genus [75] – nowa- dayscalledMumfordcurves–aswellastoabelianvarietieswithsplittorusreduc- tion[76].Moreover,Mumford’sconstructionsevenworkovercompleteNoetherian ringsofhigherdimension. The relationship between formal algebraic geometry and rigid geometry was clarified by Raynaud in [80]. As a sort of reverse, Raynaud worked on the rigid analytic uniformizationof abelian varieties and their duals over non-Archimedean fields[79]. The ideas of Mumford and Raynaud were picked up by Chai and Faltings and generalizedtoabelianvarietieswithsemi-abelianreductionsoverfieldsoffractions of complete Noetherian normal rings of higher dimension. Whereas in the rigid analytic context, the periods of the uniformization enter the scene quite naturally evenintheabsenceofapolarization,ChaiandFaltingsmadetheobservationthatthe periodsareencodedinthecoefficientsofthethetafunctionassociatedtoaprincipal polarization, in analogy to the complex case. So, for them it was not necessary to invokerigidgeometry. Nevertheless, rigid geometry is a means to unfold the geometric ideas behind theformalconstructionsusedbyMumford,ChaiandFaltings.Theresultsonuni- formization and construction provide a method to parameterize polarized abelian varieties and their semi-abelian degeneration in a universal way. So, they became the essential ingredients for the construction of a toroidal compactification of the modulispaceofpolarizedabelianvarietiesbyChaiandFaltings;cf.[27]. This book thoroughlytreats the main results on rigid geometryand their appli- cations as they grew out of the notes of Tate. The focus of this book lies on the arithmeticgeometryofcurvesandtheirJacobiansovernon-Archimedeanfields. After an introduction to rigid geometry in Chap. 1, we directly concentrate on the main topic. Following ideas of Drinfeld and Manin [64], Mumford curves are treated in Chap. 2 via classical Schottky uniformization. Their Jacobians are rigid analytictoriwhichareconstructedbyautomorphicfunctions.Thisisexplainedon an elementary level. Thus, we achieve the rigid analytic counterpart of the fasci- natingtheoryofRiemannsurfacesandtheirJacobians.Theremainderofthebook (Chaps. 3 to 7) deals with smooth rigid analytic curves and their semi-stable re- ductionsorwithpropersmoothrigidanalyticgroupvarietiesandtheirsemi-abelian reductions.Theintentionhereistocomprehensivelypresenttherigidanalyticuni- formization and construction of curves and their Jacobians or of abelian varieties over non-Archimedean fields. Moreover, the structure of abeloid varieties, which arethecounterpartsofcompactcomplexLiegroups,ispresentedindetails. Thereaderisassumedtobefamiliarwithbasicalgebraicgeometryinthestyleof Grothendieckandwithstandardfactsaboutabelianvarieties.Thereadercanconsult [15,Chaps.2and9],[60]and[74]. Since there are several books which deal with the foundations of rigid geome- try, cf. [1, 9, 10], there is no need to develop it again. Therefore, the prerequisites Preface ix onclassicalrigidgeometryareonlysurveyedinChap.1withoutgivingproofs.In the same way the basic results on the relation between formal and rigid geome- try are handled in Chap. 3, as they are presented in [14] and were revisited a few yearsagoin[1].Forthebasictheoryofformalandrigidgeometrythereadermay alsoconsult[9]whereitiscarefullyexplained.Thereareotherfoundationsofnon- ArchimedeananalysisbyBerkovich[6]andHuber[47],butthesearenotinvolved in this book. So, we concentrate on the main applications which are not touched oronlypartiallystudiedinotherbooks;cf.[30]and[35].Comparedtotheexisting literature,manyproofshavebeensubstantiallyimprovedandsomenewresultshave beenadded. It is a pleasure for me to express my gratitude to my students Sophie Schmieg andAlexMorozovforproofreadingandcomments.AlsoIwouldliketothankcol- leagues,includingSiegfriedBosch,BarryGreen,UrsHartl,DinoLorenzini,Florian Pop, Stefan Wewers, for discussions and valuable suggestions. I am especially in- debtedtoErnstKani,whohelpedmetoeditthemanuscript. InparticularIamgladtoacknowledgeheretheextraordinaryhelpfromMichel Raynaud,whocontributedmanyideastothisbook. Münster,Germany WernerLütkebohmert September2015 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 ClassicalRigidGeometry . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Non-ArchimedeanFields . . . . . . . . . . . . . . . . . . . . . 1 1.2 RestrictedPowerSeries . . . . . . . . . . . . . . . . . . . . . . 3 1.3 AffinoidSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 TheMaximumPrinciple . . . . . . . . . . . . . . . . . . . . . . 10 1.5 RigidAnalyticSpaces . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 CoherentSheaves . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 LineBundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8 AlgebraizationofProperRigidAnalyticCurves . . . . . . . . . 26 2 MumfordCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 Tate’sEllipticCurve . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 SchottkyGroups . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 DefinitionandProperties . . . . . . . . . . . . . . . . . . . . . 49 2.4 Skeletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5 AutomorphicFunctions . . . . . . . . . . . . . . . . . . . . . . 62 2.6 Drinfeld’sPolarization . . . . . . . . . . . . . . . . . . . . . . . 69 2.7 RigidAnalyticToriandTheirDuals . . . . . . . . . . . . . . . 73 2.8 JacobianVarietyofaMumfordCurve . . . . . . . . . . . . . . . 83 2.9 Riemann’sVanishingTheorem . . . . . . . . . . . . . . . . . . 91 3 FormalandRigidGeometry . . . . . . . . . . . . . . . . . . . . . . 103 3.1 CanonicalReductionofAffinoidDomains . . . . . . . . . . . . 104 3.1.1 FunctorsA (cid:2)A˚ andA (cid:2)A(cid:2) . . . . . . . . . . . . 104 K K K K 3.1.2 FormalAnalyticSpaces . . . . . . . . . . . . . . . . . . 106 3.1.3 FinitenessTheoremofGrauert-Remmert-Gruson . . . . . 111 3.2 AdmissibleFormalSchemes. . . . . . . . . . . . . . . . . . . . 113 3.3 GenericFiberofAdmissibleFormalSchemes . . . . . . . . . . 117 3.4 ReducedFiberTheorem . . . . . . . . . . . . . . . . . . . . . . 123 3.4.1 AnalyticMethodofGrauert-Remmert-Gruson . . . . . . 124 xi xii Contents 3.4.2 ElementaryMethodofEpp . . . . . . . . . . . . . . . . 126 3.4.3 TheNaturalApproach . . . . . . . . . . . . . . . . . . . 128 3.5 ComplementsonFlatness . . . . . . . . . . . . . . . . . . . . . 149 3.6 ApproximationinSmoothRigidSpaces . . . . . . . . . . . . . 155 3.7 CompactificationofSmoothCurveFibrations . . . . . . . . . . 169 4 RigidAnalyticCurves . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.1 FormalFibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.2 GenusFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.3 MeromorphicFunctions . . . . . . . . . . . . . . . . . . . . . . 196 4.4 FormalStableReduction. . . . . . . . . . . . . . . . . . . . . . 201 4.5 StableReduction . . . . . . . . . . . . . . . . . . . . . . . . . . 210 4.6 UniversalCoveringofaCurve. . . . . . . . . . . . . . . . . . . 212 4.7 CharacterizationofMumfordCurves . . . . . . . . . . . . . . . 215 5 JacobianVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.1 JacobianofaSmoothProjectiveCurve . . . . . . . . . . . . . . 218 5.2 GeneralizedJacobianofaSemi-StableCurve . . . . . . . . . . . 221 5.3 LiftingoftheJacobianoftheReduction. . . . . . . . . . . . . . 231 5.4 MorphismstoRigidAnalyticGroupswithSemi-Abelian Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.5 UniformizationofJacobians . . . . . . . . . . . . . . . . . . . . 240 5.6 ApplicationstoAbelianVarieties . . . . . . . . . . . . . . . . . 247 6 RaynaudExtensions . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1 BasicFacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.2 LineBundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 6.4 Algebraization . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.5 PolarizationofJacobians . . . . . . . . . . . . . . . . . . . . . 291 6.6 ParameterizingDegeneratingAbelianVarieties . . . . . . . . . . 303 7 AbeloidVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7.1 BasicFactsonAbeloidVarieties . . . . . . . . . . . . . . . . . 310 7.2 GenerationofSubgroupsbySmoothCovers . . . . . . . . . . . 314 7.3 ExtensionofFormalTori . . . . . . . . . . . . . . . . . . . . . 321 7.4 MorphismsfromCurvestoGroups . . . . . . . . . . . . . . . . 326 7.5 StableReductionofRelativeCurves . . . . . . . . . . . . . . . 331 7.6 TheStructureTheorem . . . . . . . . . . . . . . . . . . . . . . 342 7.7 ProofoftheStructureTheorem . . . . . . . . . . . . . . . . . . 346 Appendix Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . 355 A.1 SomeNotionsaboutGraphs . . . . . . . . . . . . . . . . . . . . 355 A.2 TorusExtensionsofFormalAbelianSchemes . . . . . . . . . . 358 A.3 CubicalStructures . . . . . . . . . . . . . . . . . . . . . . . . . 364 GlossaryofNotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381