Author | Lang, Serge. author |
---|---|
Title | Cyclotomic Fields I and II [electronic resource] / by Serge Lang |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1990 |
Edition | Combined Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0987-4 |
Descript | XVII, 436 p. online resource |
1 Character Sums -- 1. Character Sums over Finite Fields -- 2. Stickelbergerโs 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 (ยฑ 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: Washingtonโs 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. Analytic Representation of Roots of Unity -- Appendix: Barskyโs 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 Kolyvaginโs โEuler Systemโ -- 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