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On the Galois structure of circular units in \(\mathbb{Z}_p\)-extensions. (Sur la structure galoisienne des unités circulaires dans les \(\mathbb{Z}_p\)-extensions.) (French) Zbl 0911.11051
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.

MSC:
11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R34 Galois cohomology
Software:
PARI/GP
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References:
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