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Raf Cluckers; Immanuel Halupczok
Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero
(Intégration de fonctions de classe motivique exponentielle, uniforme dans tous les corps locaux de caractéristique nulle)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 45-78, doi: 10.5802/jep.63
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Class. Math.: 14E18, 03C10, 11S80, 11Q25, 40J99
Keywords: Motivic integration, motivic Fourier transforms, motivic exponential functions, $p$-adic integration, non-archimedean geometry, Denef-Pas cell decomposition, quantifier elimination, uniformity in all local fields

Résumé - 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 [10], [11] and [5]. 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.

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