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\(p\)-adic \(L\)-functions of an abelian extension of a totally real field. (Fonctions \(L\) \(p\)-adiques d’une extension abelienne d’un corps totalement réel.) (French) Zbl 0358.12009

Groupe d’Etude d’Anal. ultrametr., 3e Annee 1975/76, Journ. d’Anal. ultrametr., Marseille-Luminy 1976, Fasc. 2, Expose J 11, 11 p. (1977).
Let \(K\) be a totally real field. An integral ideal \(f\) of \(K\) corresponds to an abelian extension \(M\supset K\) with Galois group \(G(M/K)\). Let \(\chi\) be a character of \(G(M/K)\),then one has associated partial \(\zeta\) functions \(\zeta_f(r,s)\) and \(L\)-functions \(L(s,\chi)\). An explicit calculation of \(L(1-n,\chi)\) shows that these values are rational. Moreover some related expressions can be interpolated into \(p\)-adic meromorphic functions. In this way a \(p\)-adic \(L\)-function \(L_p(s,\chi)\) is build for this situation. As consequences one finds easy proofs of known results: Coates’ congruences, Leopoldt’s formula, Kronecker’s limit formula.

MSC:

11S40 Zeta functions and \(L\)-functions
11R80 Totally real fields
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