Submission date: 8 January 2007

Abstract:

We generalize the motivic incarnation morphism from the theory of arithmetic motivic integration to the relative case, where we work over a base variety S over a field k of characteristic 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 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 an appropriate Grothendieck ring of constructible effective Chow motives. This morphism yields a motivic realization of parameterized arithmetic integrals. Finally, we define relative arc and jet spaces, and the three relative motivic Poincaré series.

Mathematics Subject Classification: 12L12, 14C15, 03C10

Keywords and phrases: pseudo-finite fields, motives, motivic integration

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