On the Galois structure of circular units in \(\mathbb{Z}_p\)-extensions
type: Item
instance of: scholarly article
title: On the Galois structure of circular units in \(\mathbb{Z}_p\)-extensions
zbMATH Open document ID: 0911.11051
author: Jean-Robert Belliard
published in: Journal of Number Theory
publication date: 1999-05-05T00:00:00Z
review text: Let \(K\) be a real abelian field and \(p\) a rational prime which does not ramify in \(K\). The aim of this paper is to study the Galois behaviour of Sinott's groups of circular units \(C_s(K_n)\) in the cyclotomic \(\mathbb{Z}_p\)-extension \(K_\infty= \bigcup_{n\geq 0}K_n\). For this purpose, the author introduces a certain submodule \(D_n(K)\) of \(p\)-units of \(K_n\) which contains only universal norms (in \(K_\infty/K_n\)) and whose intersection \(\widehat{D}_n(K)\) with the units of \(K_n\) is ``close to'' \(C_s(K_n)\). For instance, \(C_s(K)= \{\pm 1\}\times \widehat{D}_0(K)\), \(C_s(K_n)= C_s(K)\). \(\widehat{D}_n(K)\) for \(n>0\), and \(\varprojlim C_s(K_n)\otimes \mathbb{Z}_p= \varprojlim \widehat{D}_n(K)\otimes \mathbb{Z}_p\). The character of \(D_n(K) \otimes \mathbb{Q}\) is computed. Under certain restrictive hypotheses relative to norms inside group algebras of cyclotomic fields associated with the prime factors of the conductor of \(K\), the author also determines the Galois module structure of \(D_n(K)\) and, as a consequence, the cohomology of \(C_s(K_n)\) in the cyclotomic tower.
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 1207951
zbMATH Keywords: real abelian field
zbMATH Keywords: Galois behaviour
zbMATH Keywords: circular units
zbMATH Keywords: cyclotomic \(\mathbb{Z}_p\)-extension
zbMATH Keywords: universal norms
zbMATH Keywords: Galois module structure
zbMATH Keywords: cohomology
zbMATH Keywords: cyclotomic tower
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: jnth.1997.2200
OpenAlex ID: W2004761329
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cites work: Q3943047
DOI: 10.1006/JNTH.1997.2200
description: scientific article; zbMATH DE number 1207951