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, Tome 5 (2018) , pp. 45-78.

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.

Reçu le : 2016-12-29
Accepté le : 2017-11-10
Publié le : 2017-11-29
DOI : https://doi.org/10.5802/jep.63
Classification : 14E18,  03C10,  11S80,  11Q25,  40J99
Mots 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
@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'\'Ecole polytechnique --- Math\'ematiques},
     pages = {45--78},
     publisher = {\'Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.63},
     zbl = {06988573},
     mrnumber = {3732692},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2018__5__45_0/}
}
Cluckers, Raf; Halupczok, Immanuel. 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/item/JEP_2018__5__45_0/

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