The rationality of the Poincaré series associated to the p-adic points on a variety
type: Item
instance of: scholarly article
title: The rationality of the Poincaré series associated to the p-adic points on a variety
zbMATH Open document ID: 0537.12011
DOI: 10.1007/BF01389133
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143136
review text: 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.
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 3853212
zbMATH Keywords: p-adic variety
zbMATH Keywords: rationality of Poincaré series
zbMATH Keywords: rational function
zbMATH Keywords: elimination of quantifiers
author: Jan Denef
MaRDI profile type: Publication
cites work: Resolution of Singularities and Division of Distributions
cites work: Diophantine Problems Over Local Fields I
cites work: The analytic continuation of generalized functions with respect to a parameter
cites work: Meromorphic property of the functions \(P^ \lambda\)
cites work: On the Poincaré series associated to the p-adic points on a curve
cites work: Decision procedures for real and p‐adic fields
cites work: Q5531082
cites work: Rational points in Henselian discrete valuation rings
cites work: Reflective functions on p-adic fields
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Complex powers and asymptotic expansions. I. Functions of certain types.
cites work: Some Observations on Higher Degree Characters
cites work: Lectures on forms of higher degree. Notes by S. Raghavan
cites work: Sets Definable Over Finite Fields: Their Zeta-Functions
cites work: Q4080695
cites work: On definable subsets of p-adic fields
cites work: On the rationality of certain generating functions
cites work: Formally \(p\)-adic fields
cites work: Q3921318
cites work: Poles of a Two-Variable P-Adic Complex Power
cites work: Algebraic theories with definable Skolem functions
cites work: On the elementary theory of Hensel fields
cites work: Q3708785
OpenAlex ID: W2064837400
description: scientific article; zbMATH DE number 3853212