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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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