Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields

Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields

Relative Motives and the Theory of Pseudo-finite Fields

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, ...

Motivic Integration and the Grothendieek Group of Pseudo-Finite Fields

Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general typ...

Motives for perfect PAC fields with pro-cyclic Galois group

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which...

Motivic Poisson Summation

We develop a “motivic integration” version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and show (under some assumptions) that the Fourier transform of a conjugation-invariant test function does not depend on the form of the division algebra. This yields a motivi...