Motivic Integration and the Grothendieek Group of Pseudo-Finite Fields

Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields

Motivic Integration and the Grothendieck 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...

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, ...

A Trace Formula for Rigid Varieties, and Motivic Weil Generating Series for Formal Schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field of equicharacteristic zero, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the analytic germ of f at x. We develop a theory of motivic integration for formal schem...

An Overview of Arithmetic Motivic Integration Julia Gordon and Yoav Yaffe

The aim of these notes is to provide an elementary introduction to some aspects of the theory of arithmetic motivic integration, as well as a brief guide to the extensive literature on the subject. The idea of motivic integration was introduced by M. Kontsevich in 1995. It was quickly developed by J. Denef and F. Loeser in a series of papers [12], [15], [11], and by others. This theory, whose a...