Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

TL;DR

This paper extends motivic integration to all residue field characteristics for Henselian discretely valued fields of characteristic zero, providing a unified axiomatic framework and proving fundamental theorems like change of variables and Fubini.

Contribution

It introduces an axiomatic approach to motivic integration applicable in all residue characteristics, generalizing previous equicharacteristic zero results with richer angular component maps.

Findings

Proves a general change of variables formula

Establishes a general Fubini Theorem

Specializes to known motivic and p-adic integrals

Abstract

We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to previously known versions of motivic integration by e.g. the second author and J. Sebag and to classical p-adic integrals.