Definable sets, motives and \(p\)-adic integrals
type: Item
instance of: scholarly article
zbMATH Open document ID: 1040.14010
DOI: 10.1090/S0894-0347-00-00360-X
author: Jan Denef
author: François Loeser
published in: Journal of the American Mathematical Society
publication date: 2001-02-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9910107
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 03C98
Mathematics Subject Classification ID: 03C68
Mathematics Subject Classification ID: 03C95
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G27
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 12L10
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 14F42
zbMATH DE Number: 1566267
zbMATH Keywords: arithmetic geometry
zbMATH Keywords: motives
zbMATH Keywords: arithmetic motivic integration
zbMATH Keywords: field arithmetic
zbMATH Keywords: first-order model theory
zbMATH Keywords: arithmetical Poincaré series
zbMATH Keywords: zeta functions
zbMATH Keywords: rationality questions
zbMATH Keywords: Galois stratifications
zbMATH Keywords: definable subassignments
zbMATH Keywords: \(p\)-adic integration
MaRDI profile type: Publication
arXiv ID: math/9910107
cites work: The elementary theory of finite fields
cites work: On the motive of a quotient variety
cites work: On the Poincaré series associated to the p-adic points on a curve
cites work: Q3988353
cites work: Grothendieck rings of \(\mathbb{Z}\)-valued fields
cites work: La conjecture de Weil. I
cites work: The rationality of the Poincaré series associated to the p-adic points on a variety
cites work: p-adic semi-algebraic sets and cell decomposition.
cites work: On the Degree of Igusa's Local Zeta Function
cites work: Q4267947
cites work: Germs of arcs on singular algebraic varieties and motivic integration
cites work: Motivic exponential integrals and a motivic Thom-Sebastiani theorem
cites work: Q4539511
cites work: Q4052071
cites work: Galois stratification over Frobenius fields
cites work: Effective counting of the points of definable sets over finite fields
cites work: Q3027108
cites work: Solving diophantine problems over all residue class fields of a number field and all finite fields
cites work: Descent, motives and K-theory.
cites work: Sets Definable Over Finite Fields: Their Zeta-Functions
cites work: Number of Points of Varieties in Finite Fields
cites work: Rationality of p-adic Poincaré series: Uniformity in p
cites work: Reduction modulo \(p^ n\) des sous-ensembles analytiques fermés de \(\mathbb Z_p^N\).
cites work: Uniform p-adic cell decomposition and local zeta functions.
cites work: Q4297057
cites work: Q5557671
cites work: Quelques applications du théorème de densité de Chebotarev
cites work: Reduction modulo pn of p-adic subanalytic sets
OpenAlex ID: W2131242042
title: Definable sets, motives and \(p\)-adic integrals
review text: In the paper under review, the authors considerably generalize several of their previously established methods and results in arithmetic geometry, and that in various directions. Namely, in a previous paper [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, 201--232 (1999; Zbl 0928.14004)], they had developed a general theory of integration -- called motivic integration -- on the space of arcs of an algebraic variety \(X\) over a field \(k\) of characteristic zero. This theory, which they now refer to as ``geometric'' motivic integration, is here replaced by a suitable arithmetic analogue which appears to be much better adapted to the study of certain rationality questions concerning Poincaré series of arithmetic schemes. More precisely, the authors develop a different kind of motivic integration theory, the so-called ``arithmetic'' motivic integration, which is largely based on methods from general field arithmetic and first-order model theory in mathematical logic. On the other hand, the Poincaré series \(P_p(T)\) associated to any pair \((X,p)\),where \(X\) is a reduced and separated scheme of finite type over \(\mathbb{Z}\) and \(p\) is a prime number, is known to be a rational function. This fact was proved by the first author [\textit{J. Denef}, Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)]. Then, just a little later, J. Denef, J. Pas, and A. Macintyre proved, independently, that the degrees of the nominator and denominator of the rational function \(P_p(T)\) are bounded independently of the prime number \(p\). In view of these deep results, the basic task of the paper under review was to derive a much stronger uniformity result by constructing a canonical rational function \(P_{\text{ar}}(T)\) which specializes to \(P_p(T)\) for almost all prime numbers \(p\). In fact, it is precisely to this end that the authors develop their new ``arithmetic'' motivic integration theory, in the first part of the paper, which then turns out to be the right tool …
reviewed by: Werner Kleinert
description: scientific article