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Residue at \(s=1\) of \(p\)-adic zeta functions. (Résidu en \(s=1\) des fonctions zêta \(p\)-adiques.) (French) Zbl 0651.12010
Let \(F\) be a totally real degree \(n\) extension of \(\mathbb Q\) and let \(O_ F\) be its ring of integers. Let \(p\) be a prime integer. The \(p\)-adic zeta function \(\zeta_{F,{\mathfrak p}}\) was defined in different ways by Serre, Deligne and Ribet, Pierrette Cassous-Nogues and Barsky. Here the author shows that \[ \lim_{s\to 1}(s-1)\zeta_{F,{\mathfrak p}}(s)=2^nR_ phE_ p(1)/w\sqrt{D}\tag{1} \] where \(R_ p\) is the \(p\)-adic regulator, \(h\) is the class number of \(F\), \(E_ p(1)=\prod_{{\mathfrak p}| p,{\mathfrak p}\in O_ F}(1-1/N({\mathfrak p}))\) (with \(N\) the norm of \(F\) over \(\mathbb Q\), which is not specified in the article), \(w\) is the number of roots of unity and \(D\) is the discriminant of \(F\). (The equation (1) was already proven by A. Amice and J. Fresnel when \(F\) is an abelian extension in [Acta Arith. 20, 353–384 (1972; Zbl 0217.04303)]). The \(p\)-adic distribution introduced by Yvette Amice helps the author to translate P. Cassou-Nogues’ results and deepen all of them in order to obtain inequalities about the Gauss norm of certain polynomials. He then computes the residue thanks to considerations on finitely generated subgroups in \(\mathbb R^ n\).

MSC:
11S40 Zeta functions and \(L\)-functions
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11R29 Class numbers, class groups, discriminants
11R80 Totally real fields
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References:
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