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Multiplicity of the poles of the Poincaré series of a p-adic subanalytic set. (English) Zbl 0682.12013

Sémin. Théor. Nombres, Univ. Bordeaux I 1987-1988, Exp. No. 43, 8 p. (1988).
With any subset S of \({\mathbb{Z}}_ p^ m\) (where \({\mathbb{Z}}_ p\) denotes the ring of p-adic integers) one associates formally the Poincaré series \(P_ S(T)=\sum^{\infty}_{n=0}N_{S,n}T^ n \), with \(N_{S,n}\) the number of points in the reduction of \(S\quad modulo\quad n.\) The author has shown previously that \(P_ S(T)\) is a rational function of T when S is a closed analytic subset [Invent. Math. 77, 1-23 (1984; Zbl 0537.12011)] and more generally (together with L. van den Dries) when S is subanalytic [Ann. Math., II. Ser. 128, No.1, 79-138 (1988)].
In the present paper he improves, in the subanalytic case, the upper bound for the order of the poles of \(P_ S(T)\) from \(m+1\) to m by combining the methods of the two quoted papers; in particular a result from the theory of formal languages is required.
Reviewer: F.Herrlich

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12J25 Non-Archimedean valued fields
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions

Citations:

Zbl 0537.12011
Full Text: EuDML