## 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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