RÉSUMÉ

In the theory of arithmetic motivic integration, a central role is played by a motivic incarnation morphism $chi$ from the Grothendieck ring $K_0$ of the theory of pseudo-finite fields, over some field $k$ of characteristic $0$, to the $Q$-tensor product of the Grothendieck ring of rational Chow motives over $k$. This morphism associates a virtual motive to every ring formula over $k$. When the ring formula is quantifier-free, it defines a constructible subset of some affine space over $k$, and its image will be exactly the motive of this subset. To give a non-trivial example: the image of the formula "there exists a non-zero element $y$ such that $x=y^2$" over $Q$ ,will be $(L-1)/2$, where $L$ is the Lefschetz motive. The key to the construction of the morphism $chi$ is Fried and Jarden's quantifier elimination over pseudo-finite fields, which yields a set of generators for $K_0$ with a nice geometric interpretation. Unfortunately, their proof is not very geometric. We will extend the theory to the relative case, working over a base variety $S/k$, using a relative adaptation of Voevodsky's formalism to replace the Chow motives. As an intermediate result, we obtain a theory of relative motives, associating to any $S$-variety $X$ its motives with and without proper support, and we obtain a character decomposition of these motives if a finite group $G$ acts on $X$, with some nice Frobenius properties. Our main tools are Bittner's presentation of the relative Grothendieck ring, and Guillen and Navarro-Aznar's criterion for the extension of homology functors. Furthermore, we give a geometric proof of quantifier elimination for pseudo-finite fields, in the relative setting, which, we hope, will make the theory more appealing to geometers. To conclude, we construct a relative motivic incarnation morphism $chi$. This morphism provides a motivic specialization of arithmetic motivic integrals with parameters, a work in progress by Cluckers and Loeser.

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