CYCLOTOMIC UNITS IN ZP-EXTENSIONS

Abstract

Let K0 be the maximal real subfield of the field generated by the p-th root of 1 over Q, and K(infinity) be the basic Z(p)-extension of K0 for a fixed odd prime p. Let K(n) be its n-th layer of this tower. For each n, we denote the Sylow p-subgroup of the ideal class group of K(n) by A(n), and that of E(n)/C(n) by B(n), where E(n) (resp. C(n)) is the group of units (resp. cyclotomic units of K(n). In section 2 of this paper, we describe structures of the direct and inverse limits of B(n). The direct limit, in particular, is shown to be a direct sum of lambda-copies of p-divisible groups and a finite group M, where lambda is the Iwasawa lambda-invariant for K(infinity) over K0. In section 3, we prove that the capitulation of A(n) in A(m) is isomorphic to M for m much greater than n much greater than 0 by using cohomological arguments. Hence if we assume Greenberg's conjecture (lambda = 0), then A(n) is isomorphic to B(n) for n much greater than 0.