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On the cyclotomic unit group and the ideal class group of a real abelian number field. II. (English) Zbl 0879.11059
The paper under review is a continuation of the author’s previous work [J. Number Theory 64, 211-222 (1997; see Zbl 0879.11060 above)]. Let $$p$$ be a fixed odd prime number and $$K$$ a real abelian number field. For the cyclotomic $$\mathbb{Z}_p$$-extension $$K_\infty$$ of $$K$$, denote by $$K_n$$ the $$n$$th layer of $$K_\infty/K$$, by $$E_n$$ the group of units in $$K_n$$, and by $$C_n$$ the group of cyclotomic units in $$K_n$$ in the sense of W. Sinnott [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. Let $$A_n$$ and $$B_n$$ be the $$p$$-Sylow subgroups of the ideal class group of $$K_n$$ and the quotient group $$E_n/C_n$$, respectively. In the previous paper cited above, the author proved the following: Assume that $$K$$ satisfies either (i) the conductor of $$k$$ is $$p$$, or (ii) $$K$$ has degree prime to $$p$$ and $$p$$ remains prime in $$K$$. If Greenberg’s conjecture is valid, then $$A_n\simeq B_n$$ as Galois modules for sufficiently large $$n$$. Here Greenberg’s conjecture states that $$\#A_n$$ remains bounded as $$n\to \infty$$ for any totally real number field [R. Greenberg, Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)].
In the paper under review, assume that $$K$$ satisfies either (i) or (ii) as above. Let $${\mathcal A}$$ and $${\mathcal B}$$ be the projective limits of $$A_n$$ and $$B_n$$ with respect to the norm maps, respectively. Denote by $${\mathcal A}_{\text{tor}}$$ the submodule of all $$\mathbb{Z}_p$$-torsion elements in $${\mathcal A}$$. Put $$\Delta= \text{Gal} (K/ \mathbb{Q})$$ and $$\Gamma_n = \text{Gal} (K_\infty/K_n)$$. Let $$\chi\in \operatorname{Hom} (\Delta,\overline \mathbb{Q}^\times_p)$$. For a $$\mathbb{Z}_p [\Delta]$$-module $$M$$, denote by $$M^\chi$$ the $$\chi$$-component of $$M$$. For a $$\Gamma_n$$-module $$M$$, denote by $$M_{\Gamma_n}$$ the $$\Gamma_n$$-coinvariant part of $$M$$ and by $$M^{\Gamma_n}$$ the $$\Gamma_n$$-invariant part of $$M$$. First, he shows as a key theorem that $$0\to ({\mathcal A}_{\text{tor}})_{\Gamma_n} \to A_n \to ({\mathcal A}/{\mathcal A}_{\text{tor}})_{ \Gamma_n} \to 0$$ and $$0\to {\mathcal B}_{\Gamma_n} \to B_n \to ({\mathcal A}_{\text{tor}})^{\Gamma_n} \to 0$$ are exact sequences of Galois modules for all $$n\geq 0$$. Next, using this, he gives two sufficient conditions, without assuming Greenberg’s conjecture, that $$A_n^\chi \simeq B^\chi_n$$ as Galois modules for sufficiently large $$n$$. One of these is that if we further assume that $$\# A^\chi_0 =p$$, then $$A^\chi_n \simeq B^\chi_n$$ as Galois modules for sufficiently large $$n$$. This result and the previous result cited above enable us to conclude that $$A_n \simeq B_n$$ as Galois modules for sufficiently large $$n$$ in the case where $$p=3$$ and $$K= \mathbb{Q} (\sqrt m)$$ in which $$p =3$$ remains prime, $$1\leq m\leq 10,000$$
Furthermore, he investigates the structures of $$A^\chi_n$$ and $$B^\chi_n$$ as $$\mathbb{Z}_p$$-modules for sufficiently large $$n$$ using the two exact sequences above. Let $$B_\infty$$ be the inductive limit of $$B_n$$ with respect to the natural inclusion maps. Then he gives a beautiful result that if $$K$$ satisfies either (i) or (ii) as above, then $${\mathcal A}^\chi \simeq \widehat {B^\chi_\infty}$$ as $$\mathbb{Z}_p$$-modules, where $$\widehat {B^\chi_\infty}$$ is the Pontryagin dual of $$B^\chi_\infty$$. He also shows that under the same assumption that the $$p$$-rank of $$A^\chi_n$$ is equal to that of $$B^\chi_n$$ for sufficiently large $$n$$.
Reviewer: H.Taya (Sendai)

##### MSC:
 11R18 Cyclotomic extensions 11R27 Units and factorization 11R23 Iwasawa theory 11R37 Class field theory 11R20 Other abelian and metabelian extensions
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##### References:
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