Abstract
LetK0 be the maximal real subfield of the field generated by thep-th root of 1 over ℚ, andK∞ be the basic Zp-extension ofK0 for a fixed odd primep. LetKn be itsn-th layer of this tower. For eachn, we denote the Sylowp-subgroup of the ideal class group ofKn byAn, and that ofEnCn byBn, whereEn (resp.Cn) is the group of units (resp. cyclotomic units ofKn. In section 2 of this paper, we describe structures of the direct and inverse limits ofBn. The direct limit, in particular, is shown to be a direct sum of λ copies ofp-divisible groups and a finite group M, where λ is the Iwasawa λ-invariant for K∞ overK0. In section 3, we prove that the capitulation ofAn inAm is isomorphic to M form ≫n ≫ 0 by using cohomological arguments. Hence if we assume Greenberg’s conjecture (λ = 0), thenAn is isomorphic toBn forn ≫ 0.
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