Journal Article

Relative Motives and the Theory of Pseudo-finite Fields

Johannes Nicaise

in International Mathematics Research Papers

Volume 2007, issue Published in print January 2007 | ISSN: 1687-3017
Published online January 2007 | e-ISSN: 1687-3009 | DOI: https://dx.doi.org/10.1093/imrp/rpm001
Relative Motives and the Theory of Pseudo-finite Fields

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We generalize the motivic incarnation morphism from the theory of arithmetic 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 relative 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 [math] with the 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.

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Subjects: Mathematics

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