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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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