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Motivic Igusa zeta functions
We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to theExpand
Geometry on Arc Spaces of Algebraic Varieties
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants andExpand
Germs of arcs on singular algebraic varieties and motivic integration
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of theExpand
p-adic semi-algebraic sets and cell decomposition.
L 1. Notation. Let p denote a fixed prime number, Zp the ring of p-adic integers and Qp the field of p-adic numbers. Let K be a fixed finite field extension of Qp. For χ € K let ord χ e Z u {+ 00}Expand
Weights of exponential sums, intersection cohomology, and Newton polyhedra
(1.1) Throughout this paper k always denotes a finite field Fq with q elements, and E a prime number not dividing q. The algebraic closure of a field K is denoted by / ( . Let ~b: k--+ C • be aExpand
Report on Igusa's local zeta function
Lefschetz numbers of iterates of the monodromy and truncated arcs
Abstract We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. We alsoExpand