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Semi-local units modulo cyclotomic units. (English) Zbl 0948.11042
This paper studies the Galois properties of the quotient of the group of semi-local units by its subgroup of cyclotomic units along the $${\mathbb Z}_p$$-cyclotomic extension. Let $$K$$ be an abelian extension of $$\mathbb Q$$ and $$G$$ its Galois group containing the $$p$$-roots of 1. Let $$\psi$$ an irreducible character $$G$$. The author defines for each $$Z_p[G]$$-module $$M$$ a $$\psi$$-part $$M^\psi$$ and a $$\psi$$-quotient $$M_\psi$$. Let $$K_\infty$$ the cyclotomic $$\mathbb Z_p$$ extension of $$K$$; let us denote by $$C$$ (resp. $$U$$) the inverse limit of the semi-local units (resp. the cyclotomic units). The $$p$$-adic $$L$$-function associated to $$\psi$$ can be expressed by an Iwasawa series $$g_\psi$$. The main results of this paper gives the structure of $$U^\psi/C^\psi$$ and $$(U/C)_\psi$$ by comparing them with a quotient $$\Lambda_\psi/(g_\psi)$$. The statement is easier in the case of $$U^\psi/C^\psi$$. These results precise C. Greither’s one [Ann. Inst Fourier 42, 449-499 (1991; Zbl 0757.11039)] which treated the same problem but after extending scalars of the Galois modules with $$\mathbb Q_p$$. The main tool is again Coleman’s series.

##### MSC:
 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R23 Iwasawa theory 11S23 Integral representations
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##### References:
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