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.
Par une cascade de généralisations, nous développons une théorie de l’intégration motivique qui fonctionne uniformément dans tous les corps locaux non archimédiens de caractéristique nulle, en surmontant des difficultés reliées à la ramification et à la caractéristique résiduelle petite. Nous définissons une classe de fonctions – appelées fonctions de classe motivique exponentielle – dont nous démontrons qu’elle est stable par intégration et par transformation de Fourier, étendant des résultats et des définitions de [10], [11] et [5]. Nous démontrons des résultats uniformes reliés à la rationalité et à différents types de lieux. Un ingrédient clef est une forme raffinée de l’élimination des quantificateurs de Denef-Pas, qui nous permet de comprendre des ensembles définissables dans le groupe de valeur et dans le corps valué.
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@article{JEP_2018__5__45_0,
author = {Raf Cluckers and Immanuel Halupczok},
title = {Integration of functions of~motivic~exponential~class, uniform~in~all~non-archimedean local fields of characteristic zero},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {45--78},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.63},
zbl = {06988573},
mrnumber = {3732692},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.63/}
}
TY - JOUR AU - Raf Cluckers AU - Immanuel Halupczok TI - Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 45 EP - 78 VL - 5 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.63/ DO - 10.5802/jep.63 LA - en ID - JEP_2018__5__45_0 ER -
%0 Journal Article %A Raf Cluckers %A Immanuel Halupczok %T Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 45-78 %V 5 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.63/ %R 10.5802/jep.63 %G en %F JEP_2018__5__45_0
Raf Cluckers; Immanuel Halupczok. Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 45-78. doi : 10.5802/jep.63. https://jep.centre-mersenne.org/articles/10.5802/jep.63/
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