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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.

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R23 Iwasawa theory
11S23 Integral representations
Full Text: DOI
[1] Coleman, R., Division values in local fields, Invent. math., 53, 96-116, (1979) · Zbl 0429.12010
[2] Coleman, R., Local units modulo circular units, Proc. amer. math. soc., 89, 1-7, (1983) · Zbl 0528.12005
[3] Ferrero, B.; Washington, L.C., The Iwasawa invariant μp vanishes for abelian number fields, Ann. of math., 109, 377-395, (1979) · Zbl 0443.12001
[4] Gillard, R., Unités cyclotomiques, unités semi-locales et \(Z\)_l-extensions II, Ann. inst. Fourier (Grenoble), 29, 1-15, (1979) · Zbl 0403.12006
[5] Greither, C., Class groups of abelian fields, and the main conjecture, Ann. inst. Fourier (Grenoble), 42, 449-499, (1992) · Zbl 0729.11053
[6] Iwasawa, K., On some modules in the theory of cyclotomic fields, J. math. soc. Japan, 16, 42-82, (1964) · Zbl 0125.29207
[7] Perrin-Riou, B., La fonction L p-adique de kubota et leopoldt, Arithmetic geometry, Tempe, AZ, 1993, Contemporary math., 174, (1994), ProvidenceAmerican Mathematical Society, p. 65-93 · Zbl 0836.11020
[8] Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. math., 62, 181-234, (1980) · Zbl 0465.12001
[9] Sinnott, W., On p-adic L-functions and the Riemann-Hurwitz genus formula, Compositio math., 53, 3-17, (1984) · Zbl 0545.12011
[10] Solomon, D., On the classgroups of imaginary abelian fields, Ann. inst. Fourier (Grenoble), 40, 467-492, (1990) · Zbl 0694.12004
[11] Washington, L.C., Introduction to cyclotomic fields, Graduate texts in math., 83, (1982), Springer-Verlag Berlin/New York
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