Cyclotomic Fields II

This second volume incorporates a number of results which were discovered and/or systematized since the first volume was being written. Again, I limit myself to the cyclotomic fields proper without introducing modular func tions. As in the first volume, the main concern is with class number formulas, Gauss sums, and the like. We begin with the Ferrero-Washington theorems, proving Iwasawa's conjecture that the p-primary part of the ideal class group in the cyclotomic Zp-extension of a cyclotomic field grows linearly rather than exponentially. This is first done for the minus part (the minus referring, as usual, to the eigenspace for complex conjugation), and then it follows for the plus part because of results bounding the plus part in terms of the minus part. Kummer had already proved such results (e.g. if p, (h; then p, (h;). These are now formulated in ways applicable to the Iwasawa invariants, following Iwasawa himself. After that we do what amounts to " Dwork theory," to derive the Gross Koblitz formula expressing Gauss sums in terms of the p-adic gamma function. This lifts Stickel berger's theorem p-adically. Half of the proof relies on a course of Katz, who had first obtained Gauss sums as limits of certain factorials, and thought of using Washnitzer-Monsky cohomology to prove the Gross-Koblitz formula Front Matter....Pages i-xi Measures and Iwasawa Power Series....Pages 1-25 The Ferrero-Washington Theorems....Pages 26-36 Measures in the Composite Case....Pages 37-51 Divisibility of Ideal Class Numbers....Pages 52-70 p -adic Preliminaries....Pages 71-85 The Gamma Function and Gauss Sums....Pages 86-116 Gauss Sums and the Artin-Schreier Curve....Pages 117-137 Gauss Sums as Distributions....Pages 138-153 Back Matter....Pages 155-164