## 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

### Citations:

Zbl 0186.09103; Zbl 0157.10302
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