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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
PARI/GP
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##### References:
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