Rigid Geometry of Curves and Their Jacobians

398 Pages·2016·2.11 MB·English

by  Werner Lütkebohmert

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:
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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
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MathematicsSubjectClassification(2010): 14G22,14H40,14K15,30G06
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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