Cyclotomic Fields I and II

1 Character Sums. - 1. Character Sums over Finite Fields. - 2. Stickelberger's Theorem. - 3. Relations in the Ideal Classes. - 4. Jacobi Sums as Hecke Characters. - 5. Gauss Sums over Extension Fields. - 6. Application to the Fermat Curve. - 2 Stickelberger Ideals and Bernoulli Distributions. - 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. - 3 Complex Analytic Class Number Formulas. - 1. Gauss Sums on Z/mZ. - 2. Primitive L-series. - 3. Decomposition of L-series. - 4. The ( 1)-eigenspaces. - 5. Cyclotomic Units. - 6. The Dedekind Determinant. - 7. Bounds for Class Numbers. - 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 Leopoldt Transform. - 6. The p-adic Leopoldt Transform. - 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. - 6 Kummer Theory over Cyclotomic Zp-extensions. - 1. The Cyclotomic 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-Leopoldt Theorem and the Kummer-Vandiver Conjecture. - 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 A. - 4. The Coates-Wiles Homomorphism. - 5. The Closure of the Cyclotomic Units. - 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. - 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 ? = xn. - 5. The Main Theorem for the Symbol ?x xn?n. - 6. The Main Theorem for Divisible x and ? = unit. - 7. End of the Proof of the Main Theorems. - 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 l Prime to p: Washington's Theorem. - 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. - 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(1 x) in the Composite Case. - 13 Divisibility of Ideal Class Numbers. - 1. Iwasawa Invariants in Zp-extensions. - 2. CM Fields Real Subfields and Rank Inequalities. - 3. The l-primary Part in an Extension of Degree Prime to l. - 4. A Relation between Certain Invariants in a Cyclic Extension. - 5. Examples of Iwasawa. - 6. A Lemma of Kummer. - 14 P-adic Preliminaries. - 1. The p-adic Gamma Function. - 2. The Artin-Hasse Power Series. - 3. AnalyticRepresentation of Roots of Unity. - Appendix: Barsky's Existence Proof for the p-adic Gamma Function. - 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. - 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. - 17 Gauss Sums as Distributions. - 1. The Universal Distribution. - 2. The Gauss Sums as Universal Distributions. - 3. The L-function at s = 0. - 4. The p-adic Partial Zeta Function. - Appendix by Karl Rubin. - The Main Conjecture. - 1. Setting and Notation. - 2. Properties of Kolyvagin's Euler System. - 3. An Application of the Chebotarev Theorem. - 5. The Main Conjecture. - 6. Tools from Iwasawa Theory. - 7. Proof of Theorem 5. 1. - 8. Other Formulations and Consequences of the Main Conjecture. Language: English