[Intégration de fonctions de classe motivique exponentielle, uniforme dans tous les corps locaux de caractéristique nulle]
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.
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.
Accepté le :
Publié le :
DOI : 10.5802/jep.63
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
Mot clés : Intégration motivique, transformation de Fourier motivique, fonctions motiviques exponentielles, intégration $p$-adique, géométrie non archimédienne, décomposition cellulaire de Denef-Pas, élimination des quantificateurs, uniformité dans tous les corps locaux
Raf Cluckers 1 ; Immanuel Halupczok 2
CC-BY-ND 4.0
@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, Tome 5 (2018), pp. 45-78. doi : 10.5802/jep.63. https://jep.centre-mersenne.org/articles/10.5802/jep.63/
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