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Fonction zêta \(p\)-adique des corps de nombres abéliens réels. (French) Zbl 0217.04303

We give here a new definition of the \(p\)-adic \(L\)-functions of an abelian real number field \(K\). Let \(\chi\) be a primitive character of \(K\) and \(H_\chi(m,T) = \sum_{n=1}^\infty \chi(n)n^nT^n/n\). If \(m\) is a positive integer, \(H_\chi(m,T)\) is a rational function of \(T\), without pole at \(T=1\), (provided the conductor \(f(\chi)\) does not equal \(p\)), satisfying \[ H_\chi(m, 1) = - B^m(x)/m = L(1-m, \chi). \tag{1} \] We show that for each \(m\) in \(\mathbb Z_p\) (provided the conductor \(f(\chi)\) does not equal \(p\)), \(H_\chi(m,T)\) can be continued analytically, in Krasner’s sense, on a domain containing \(T=1\). Then the values at \(T=1\) satisfy \(H_\chi(1-m, 1) = L_p (1 -m,\chi)\) for all \(m\) in \(\mathbb Z_p\). As this value depends continuously on the \(p\)-adic integer \(m\), and satisfies (1) it coincides on \(\mathbb Z_p\) with the usual \(L\)-function as defined by Leopoldt [T. Kubota and H. W. Leopoldt, J. Reine Angew. Math. 214/215, 328–339 (1964; Zbl 0186.09103)] and J. Fresnel [Ann. Inst. Fourier 17 (1967), No. 2, 281–333 (1968; Zbl 0157.10302)]. When the conductor \(f(\chi)\) is equal to \(p\) we must change \(H\) into a slightly more complicated function, but we find an analogous analytic continuation. In both cases, by means of this new definition of \(L_p\), we can prove the \(p\)-adic analytic residues formula \[ 2^{n-1} h(R_pd^{-1/2}) = \prod _{p\mid \mathfrak p}(1 - N(\mathfrak p)^{-1}) \lim_{s\to 1} (s - 1) Z_p(s,K)\] where \(n\) is the degree, \(d\) the discriminant, \(R_p\) the \(p\)-adic regulator, \(h\) the number of ideal classes and \(Z_p\) the \(p\)-adic Zeta function of the abelian field \(K\).

MSC:

11S40 Zeta functions and \(L\)-functions
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