分圆域Ⅰ和Ⅱ

Notation

Introduction

CHAPTER1CharacterSums

1.CharacterSumsoverFiniteFields

2.StickelbergersTheorem

3.RelationsintheIdealClasses

4.JacobiSumsasHeckeCharacters

5.GaussSumsoverExtensionFields

6.ApplicationtotheFermatCurve

CHAPTER2StickelbergerIdealsandBernoulliDistribution

1.TheIndexoftheFirstStickelbergerIdeal

2.BernoulliNumbers

3.IntegralStickelbergerIdeals

4.GeneralCommentsonIndices

5.TheIndexforkEven

6.TheIndexforkOdd

7.TwistingsandStickelbergerIdeals

8.StickelbergerElementsasDistributions

9.UniversalDistributions

10.TheDavenport-HasseDistribution

Appendix.Distributions

CHAPTER3ComplexAnalyticClassNumberFormulas

1.GaussSumsonZ/raZ

2.PrimitiveL-series

3.DecompositionofL-series

4.The(±I)-eigenspaces

5.CyclotomicUnits

6.TheDedekindDeterminant

7.BoundsforClassNumbers

CHAPTER4Thep-adicL-function

1.MeasuresandPowerSeries

2.OperationsonMeasuresandPowerSeries

3.TheMellinTransformandp-adicL-functionAppendix.Thep-adicLogarithm

4.Thep-adicRegulator

5.TheFormalLeopoidtTransform

6.Thep-adicLeopoldtTransform

CHAPTER5IwasawaTheoryandIdealClassGroups

1.TheIwasawaAlgebra

2.WeierstrassPreparationTheorem

3.ModulesoverZp[[X]]

4.Zp-extensionsandIdealClassGroups

5.TheMaximalp-abelianp-ramifiedExtension

6.TheGaloisGroupasModuleovertheIwasawaAlgebra

CHAPTER6KummerTheoryoverCyclotomicZp-extensions

1.TheCyciotomicZp-extension

2.TheMaximalp-abelianp-ramifiedExtensionoftheCyclotomicZp-extension

3.CyclotomicUnitsasaUniversalDistribution

4.TheIwasawa-LeopoidtTheoremandtheKummer-VandiverConjecture

CHAPTER7IwasawaTheoryofLocalUnits

1.TheKummer-TakagiExponents

2.ProjectiveLimitoftheUnitGroups

3.ABasisforU(x)overΛ

4.TheCoates-WilesHomomorphism

5.TheClosureoftheCyclotomicUnits

CHAPTER8Lubin-TateTheory

1.Lubin-TateGroups

2.Formalp-adicMultiplication

3.ChangingthePrime

4.TheReciprocityLaw

5.TheKummerPairing

6.TheLogarithm

7.ApplicationoftheLogarithmtotheLocalSymbol

CHAPTER9ExplicitReciprocityLaws

1.StatementoftheReciprocityLaws

2.TheLogarithmicDerivative

3.ALocalPairingwiththeLogarithmicDerivative

4.TheMainLemmaforHighlyDivisiblexandα=xn

5.TheMainTheoremfortheSymboln

6.TheMainTheoremforDivisiblexandα=unit

7.EndoftheProofoftheMainTheorems

CHAPTER10MeasuresandIwasawaPowerSeries

1.IwasawaInvariantsforMeasures

2.ApplicationtotheBernoulliDistributions

3.ClassNumbersasProductsofBernoulliNumbersAppendixbyL.Washington:Probabilities

4.Divisibilityby!Primetop:WashingtonsTheorem

CHAPTER11TheFerrero-WashingtonTheorems

1.BasicLemmaandApplications

2.EquidistributionandNormalFamilies

3.AnApproximationLemma

4.ProofoftheBasicLemma

CHAPTER12MeasuresintheCompositeCase

1.MeasuresandPowerSeriesintheCompositeCase

2.TheAssociatedAnalyticFunctionontheFormalMultiplicativeGroup

3.ComputationofLp(l,X)intheCompositeCase

CHAPTER13DivisibilityofIdealClassNumbers

I.IwasawaInvariantsinZp-extensions

2.CMFields,RealSubfields,andRankInequalities

3.The/-primaryPartinanExtensionofDegreePrimetol

4.ARelationbetweenCertainInvariantsinaCyclicExtension

5.Examplesoflwasawa

6.ALemmaofKummer

CHAPTER14p-adicPreliminaries

I.Thep-adicGammaFunction

2.TheArtin-HassePowerSeries

3.AnalyticRepresentationofRootsofUnity

Appendix:BarskysExistenceProofforthep-adicGammaFunction

CHAPTER15TheGammaFunctionandGaussSums

1.TheBasicSpaces

2.TheFrobeniusEndomorphism

3.TheDworkTraceFormulaandGaussSums

4.EigenvaluesoftheFrobeniusEndomorphismandthep-adicGammaFunction

5.p-adicBanachSpaces

CHAPTER16GaussSumsandtheArtin-SchreierCurve

1.PowerSerieswithGrowthConditions

2.TheArtin-SchreierEquation

3.Washnitzer-MonskyCohomology

4.TheFrobeniusEndomorphism

CHAPTER17GaussSumsasDistributions

1.TheUniversalDistribution

2.TheGaussSumsasUniversalDistributions

3.TheL-functionats=0

4.Thep-adicPartialZetaFunction

APPENDIXBYKARLRUBIN

TheMainConjecture

Introduction

1.SettingandNotation

2.PropertiesofKolyvagins"EulerSystem"

3.AnApplicationoftheChebotarevTheorem

4.Example:TheIdealClassGroupofQ(μp)+

5.TheMainConjecture

6.ToolsfromIwasawaTheory

7.ProofofTheorem5.1

8.OtherFormulationsandConsequencesoftheMainConjecture

Bibliography

Index

　　KummersworkoncyclotomicfieldspavedthewayforthedevelopmentofalgebraicnumbertheoryingeneralbyDedekind,Weber,Hensel,Hilbert,Takagi,Artinandothers.However,thesuccessofthisgeneraltheoryhastendedtoobscurespecialfactsprovedbyKummeraboutcyclotomicfieldswhichliedeeperthanthegeneraltheory.Foralongperiodinthe20thcenturythisaspectofKummersworkseemstohavebeenlargelyforgotten,exceptforafewpapers,amongwhicharethosebyPollaczek[Po],Artin-Hasse[A-H]andVandiver.Inthemid1950s,thetheoryofcyclotomicfieldswastakenupagainbyIwasawaandLeopoldt.Iwasawaviewedcyclotomicfieldsasbeinganaloguesfornumberfieldsoftheconstantfieldextensionsofalgebraicgeometry,andwroteagreatsequenceofpapersinvestigatingtowersofcyclotomicfields,andmoregenerally,GaioisextensionsofnumberfieldswhoseGaloisgroupisisomorphictotheadditivegroupofp-adicintegers.