[share_ebook] Cyclotomic Fields II (Graduate Texts in Mathematics)

Description

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 functions. 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 pxh- then pxhp+). 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 Stickelberger'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. Finally, we apply these latter results to the Ferrero-Greenberg theorem, showing that Lp'(O, X) not =0 under the appropriate conditions. We take this opportunity to introduce a technique of Washington, who defined the p-adic analogues of the Hurwitz partial zeta functions, in a way making it possible to parallel the treatment from the complex case to the p-adic case, but in a much more efficient way.