Constructible exponential functions, motivic Fourier transform and transfer principle


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.


Raf Cluckers

Katholieke Universiteit Leuven
Departement Wiskunde
Celestijnenlaan 200B
B-3001 Leuven

François Loeser

École Normale Supérieure
45, rue d’Ulm
F-75230 Paris Cedex 05