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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.) (English. French summary) Zbl 1433.14012

One motivation for motivic integration is to study \(p\)-adic integration uniformly in the local field. This is often done in a setting that provides uniformity results only for sufficiently large residue characteristic or, in the case of small residue characteristic, only when requiring bounds on ramification. One framework that does this is the model theoretic motivic integration of Cluckers and Loeser. In this paper, the authors develop a version of quantifier elimination and obtain a version of that model theoretic motivic integration that works uniformly for all non-Archimedean local fields of characteristic zero, even allowing small residue characteristic and arbitrary ramification.
The authors define “functions of motivic exponential class”. Such a function is a collection of functions \((f_K)_K\) satisfying a certain uniformity in the index \(K\), which varies over all non-Archimedean characteristic 0 local fields. They prove that this class of functions is closed under integration, which results in various uniformity results concerning \(p\)-adic integrals. For example, in the case of certain rational functions that generalize Igusa’s local zeta functions, the authors obtain a description for candidate poles that is uniform in the local field.

MSC:

14E18 Arcs and motivic integration
03C10 Quantifier elimination, model completeness, and related topics
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
40J99 Summability in abstract structures
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References:

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