doi:10.1093/imrp/rpm001 RelativeMotives and the Theory of Pseudo-finite Fields

We generalize the motivic incarnation morphism from the theory of arithmetic integra-tion to the relative case, where we work over a base variety S over a field k of charac-teristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of rela-tive quantifier elimination for pseudo-finite fields, and we construct a morphism from the Grothendieck ring of the theory of pseudo-finite fields over S, to the tensor product of Q with the Grothendieck ring of constructible effective Chow motives. This morphism yields a motivic realization of parameterized arithmetic integrals. Finally, we define rel-ative arc and jet spaces, and the three relative motivic Poincare ́ series.