Constructible exponential functions, motivic Fourier transform and transfer principle

Raf Cluckers; François Loeser

doi:http://doi.org/10.4007/annals.2010.171.1011

Abstract

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We also define spaces of motivic Schwartz-Bruhat functions on which motivic Fourier transformation induces isomorphisms. Our motivic integrals specialize to nonarchimedean integrals. We give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero, resp. of positive characteristic, having the same residue field. We also prove new results about $p$-adic integrals of exponential functions and stability of this class of functions under $p$-adic integration.

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[wal] $Go to document$ J. -L. Waldspurger, "Endoscopie et changement de caractéristique," J. Inst. Math. Jussieu, vol. 5, iss. 3, pp. 423-525, 2006.

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@article {wal, MRKEY = {2241929}, AUTHOR = {Waldspurger, J.-L.}, TITLE = {Endoscopie et changement de caractéristique}, JOURNAL = {J. Inst. Math. Jussieu}, FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu}, VOLUME = {5}, YEAR = {2006}, NUMBER = {3}, PAGES = {423--525}, ISSN = {1474-7480}, MRCLASS = {22E35 (20G25 22E60)}, MRNUMBER = {2007h:22007}, MRREVIEWER = {Bertrand Lemaire}, DOI = {10.1017/S1474748006000041}, ZBLNUMBER = {1102.22010}, }

Constructible exponential functions, motivic Fourier transform and transfer principle

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