Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero

Raf Cluckers,Immanuel Halupczok

doi:10.5802/jep.63

Abstract

Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from previous papers. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field.

Highlights

Abstract. — Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics
For K a local field with valuation ring OK with maximal ideal MK, we do not obtain new results about OK modulo the ideals nMK := {nm | m ∈ MK } for integers n > 0, but rather, we use these finite quotients as tools, in order to understand the model theory of K and the geometry of definable sets
Let us note that the use of model theory to study p-adic integrals originated in work by Denef [13], where the approach with resolution of singularities was used by Igusa in the early seventies

Summary

Introduction

1.1. — Much of the existing theory of local zeta functions and p-adic integrals has been developed for large residue field characteristic, and in the case of small residue field characteristic only with bounds on ramification. Given a definable f , the application was shown by Pas for large enough residue field characteristic [27], and for small residue field characteristic but with bounded ramification [28] Both cases treated by Pas rely on Denef-Pas quantifier elimination (the model theoretic approach). The idea is that loci of several kinds of bad behaviour are contained in proper Zariski closed subsets, uniformly in all local fields of characteristic zero, roughly as in Theorem E of [1] Many such results are already known for large enough residue field characteristic (or assuming bounds on the ramification), so that the new point is again to be completely uniform in all local fields of characteristic zero. Kowalski and the Forschungsinstitut für Mathematik (FIM) at ETH Zürich for the hospitality and invitation to R.C. to give the Nachdiplom Lectures at the ETH in 2014 related to the themes of the paper

Uniform p-adic definable sets and functions

Functions of C -class and of C exp-class

Integration of C - and C exp-functions

Quantifier elimination and related results