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The rationality of the Poincaré series associated to the p-adic points on a variety. (English) Zbl 0537.12011

Let \(f_ 1,...,f_ n\) be polynomials in m variables with coefficients in \({\mathbb{Z}}_ p\). For each \(n\geq 0\), let \(\tilde N_ n\) be the number of common zeroes of \(f_ 1,...,f_ n\) in \(({\mathbb{Z}}/p^ n{\mathbb{Z}})^ m\), and \(N_ n\) the number of such zeroes which can be lifted to zeroes of the \(f_ i\!'s\) in \({\mathbb{Z}}^ m_ p\). Igusa in the case \(n=1\) and Meuser in the general case proved that \(\tilde P(T)=\sum \tilde N_ n T^ n\) is a rational function of T. Serre asked the question whether the same holds for \(P(T)=\sum N_ n T^ n.\) This is proved in this paper using Macintyre’s theorem on the elimination of quantifiers for \({\mathbb{Q}}_ p\). Two different proofs are given, one of which has the advantage to yield a new proof of the rationality of \(\tilde P\) without using Hironaka’s resolution of singularities. The case of a two-variable series generalizing P and \(\tilde P\) is also treated.
Reviewer: J.Oesterlé

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14G20 Local ground fields in algebraic geometry
03C10 Quantifier elimination, model completeness, and related topics
12L12 Model theory of fields
13H15 Multiplicity theory and related topics
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References:

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