On the cyclotomic unit group and the ideal class group of a real abelian number field
type: Item
instance of: scholarly article
title: On the cyclotomic unit group and the ideal class group of a real abelian number field
zbMATH Open document ID: 0879.11058
author: Manabu Ozaki
published in: Journal of Number Theory
publication date: 1997-06-23T00:00:00Z
review text: Let \(p\) be a fixed odd prime number and \(K\) a real abelian number field. For the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\), denote by \(K_n\) its \(n\)th layer, 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 \textit{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. Then if \(K\) has degree prime to \(p\), then \(\# A_n= \#B_n\). Also, the Iwasawa main conjecture proved by \textit{B. Mazur} and \textit{A. Wiles} [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)] says that the characteristic ideal of the projective limit of \(A_n\) with respect to the norm maps is equal to that of \(B_n\). In the paper under review, the author considers the naturally arising problem whether \(A_n\) is isomorphic to \(B_n\) as Galois modules for sufficiently large \(n\). Put \(\Gamma_{m,n}=\text{Gal}(K_m/K_n)\) and let \(\widehat H^i (\Gamma_{m,n}, M)\) be the \(i\)th Tate cohomology group for a \(\Gamma_{m,n}\)-module \(M\). He gives fantastic results on the relations between \(A_n\) and \(B_n\). The first result is as follows: Assume that \(K\) satisfies certain conditions which are cited as condition (C) in the paper under review. Then, as Galois modules, for all \(m\geq n\geq 0\), \(\widehat H^i (\Gamma_{m,n}, A_n) \simeq \widehat H^i (\Gamma_{m,n}, B_n)\) for \(i=0\), \(-1\), and \(\widehat H^i (\Gamma_{ m,n}, A_n)\) is isomorphic to the cokernel of the natural map \(A_n\to A_m^{\Gamma_{m,n}}\), or the kernel of the natural map \(A_n\to A_m\), according as \(i=0\) or \(-1\), where \(A_m^{\Gamma_{m,n}}\) is the \(\Gamma_{m,n}\)-invariant part of \(A_m\). Next, by using this, he shows that if we further assume that Greenberg's conjecture is valid, then \(A_n \simeq B_n\) as Galois modules for sufficiently large \(n\). Here Greenberg's conjecture states that t…
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1025609
zbMATH Keywords: cyclotomic \(\mathbb{Z}_ p\)-extension
zbMATH Keywords: \(p\)-Sylow subgroups of the ideal class group
zbMATH Keywords: Galois modules
zbMATH Keywords: real abelian number field
zbMATH Keywords: group of cyclotomic units
reviewed by: Hisao Taya
MaRDI profile type: Publication
full work available at URL: jnth.1997.2086
OpenAlex ID: W2770327461
cites work: The Iwasawa invariant \(\mu_p\) vanishes for abelian number fields
cites work: On the Iwasawa Invariants of Totally Real Number Fields
cites work: Class groups of abelian fields, and the main conjecture
cites work: On relatively invariant circular units and Stickelberger elements
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cites work: Q4063245
cites work: Cohomology groups of cyclotomic units
cites work: Q4855224
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Q3995821
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: Q3943047
DOI: 10.1006/JNTH.1997.2086
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description: scientific article; zbMATH DE number 1025609