Kummer's work on cyclotomic fields paved the way for the development ofalgebraic number theory in general by Dedekind, Weber, Hensel, Hilbert,Takagi, Artin and others. However, the success of this general theory hastended to obscure special facts proved by Kummer about cyclotomic fieldswhich lie deeper than the general theory. For a long period in the 20th centurythis aspect of Kummer's work seems to have been largely forgotten, exceptfor a few papers, among which are those by Pollaczek [Po], Artin-Hasse[A-H] and Vandiver . In the mid 1950's, the theory of cyclotomic fields was taken up again byIwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analoguesfor number fields of the constant field extensions of algebraic geometry, andwrote a great sequence of papers investigating towers of cyclotomic fields,and more generally, Gaiois extensions of number fields whose Galois groupis isomorphic to the additive group ofp-adic integers.

Notation

Introduction

CHAPTER 1 Character Sums

 1.Character Sums over Finite Fields

 2.Stickelberger's Theorem

 3.Relations in the Ideal Classes

 4.Jacobi Sumsas Hecke Characters

 5.Gauss Sums over Extension Fields

 6.Application to the Fermat Curve

CHAPTER 2 Stickelberger Ideals and Bernoulli Distribution

 1.The Index of the First Stickelberger Ideal

 2.Bernoulli Numbers

 3.Integral Stickelberger Ideals

 4.General Comments on Indices

 5.The Index for k Even

 6.The Index for k Odd

 7.Twistings and Stickelberger Ideals

 8.Stickelberger Elements as Distributions

 9.Universal Distributions

 10. The Davenport-Hasse Distribution

Appendix. Distributions

CHAPTER 3 Complex Analytic Class Number Formulas

 1.Gauss Sums on Z/raZ

 2.Primitive L-series

 3.Decomposition of L-series

 4.The (±I)-eigenspaces

 5.Cyclotomic Units

 6.The Dedekind Determinant

 7.Bounds for Class Numbers

CHAPTER 4 The p-adic L-function

 1.Measures and Power Series

 2.Operations on Measures and Power Series

 3.The Mellin Transform and p-adic L-function Appendix. The p-adic Logarithm

 4.The p-adic Regulator

 5.The Formal Leopoidt Transform

 6.The p-adic Leopoldt Transform

CHAPTER 5 Iwasawa Theory and Ideal Class Groups

 1.The Iwasawa Algebra

 2.Weierstrass Preparation Theorem

 3.Modules over Zp[[X]]

 4.Zp-extensions and Ideal Class Groups

 5.The Maximal p-abelian p-ramified Extension

 6.The Galois Group as Module over the Iwasawa Algebra

CHAPTER 6 Kummer Theory over Cyclotomic Zp-extensions

 1.The Cyciotomic Zp-extension

 2.The Maximal p-abelian p-ramified Extension of the Cyclotomic Zp-extension

 3.Cyclotomic Units as a Universal Distribution

 4.The Iwasawa-Leopoidt Theorem and the Kummer-Vandiver Conjecture

CHAPTER 7 Iwasawa Theory of Local Units

 1.The Kummer-Takagi Exponents

 2.Projective Limit of the Unit Groups

 3.A Basis for U(x) over Λ

 4.The Coates-Wiles Homomorphism

 5.The Closure of the Cyclotomic Units

CHAPTER 8 Lubin-Tate Theory

 1.Lubin-Tate Groups

 2.Formal p-adic Multiplication

 3.Changing the Prime

 4.The Reciprocity Law

 5.The Kummer Pairing

 6.The Logarithm

 7.Application of the Logarithm to the Local Symbol

CHAPTER 9 Explicit Reciprocity Laws

 1.Statement of the Reciprocity Laws

 2.The Logarithmic Derivative

 3.A Local Pairing with the Logarithmic Derivative

 4.The Main Lemma for Highly Divisible x and α = xn

 5.The Main Theorem for the Symbol <x, xn>n

 6.The Main Theorem for Divisible x and α = unit

 7.End of the Proof of the Main Theorems

CHAPTER 10 Measures and Iwasawa Power Series

 1.Iwasawa Invariants for Measures

 2.Application to the Bernoulli Distributions

 3.Class Numbers as Products of Bernoulli Numbers Appendix by L. Washington: Probabilities

 4.Divisibility by ! Prime to p: Washington's Theorem

CHAPTER 11 The Ferrero-Washington Theorems

 1.Basic Lemma and Applications

 2.Equidistribution and Normal Families

 3.An Approximation Lemma

 4.Proof of the Basic Lemma

CHAPTER 12 Measures in the Composite Case

 1.Measures and Power Series in the Composite Case

 2.The Associated Analytic Function on the Formal Multiplicative Group

 3.Computation of Lp(l,X) in the Composite Case

CHAPTER 13 Divisibility of Ideal Class Numbers

 I.Iwasawa Invariants in Zp-extensions

 2.CM Fields, Real Subfields, and Rank Inequalities

 3.The/-primary Part in an Extension of Degree Prime to l

 4.A Relation between Certain Invariants in a Cyclic Extension

 5.Examples oflwasawa

 6.A Lemma of Kummer

CHAPTER 14 p-adic Preliminaries

 I.Thep-adic Gamma Function

 2.The Artin-Hasse Power Series

 3.Analytic Representation of Roots of Unity

 Appendix: Barsky's Existence Proof for the p-adic Gamma Function

CHAPTER 15 The Gamma Function and Gauss Sums

 1.The Basic Spaces

 2.The Frobenius Endomorphism

 3.The Dwork Trace Formula and Gauss Sums

 4.Eigenvalues of the Frobenius Endomorphism and the p-adic Gamma Function

 5.p-adic Banach Spaces

CHAPTER 16 Gauss Sums and the Artin-Schreier Curve

 1.Power Series with Growth Conditions

 2.The Artin-Schreier Equation

 3.Washnitzer-Monsky Cohomology

 4.The Frobenius Endomorphism

CHAPTER 17 Gauss Sums as Distributions

 1.The Universal Distribution

 2.The Gauss Sums as Universal Distributions

 3.The L-function at s = 0

 4.Thep-adic Partial Zeta Function

 APPENDIX BY KARL RUBIN

The Main Conjecture

 Introduction

 1.Setting and Notation

 2.Properties of Kolyvagin's "Euler System"

 3.An Application of the Chebotarev Theorem

 4.Example: The Ideal Class Group of Q(μp)+

 5.The Main Conjecture

 6.Tools from Iwasawa Theory

 7.Proof of Theorem 5.1

 8.Other Formulations and Consequences of the Main Conjecture

Bibliography

Index