Jan Denef

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 the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle.

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 and brings new light to some classical invariants.

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 a nontrivial additive character, and ~ , the Qt-sheaf on A~ associated to ~ and the Artin-Schreier covering t q t = x. For a morphism f : X --+ A~, with X a scheme of finite type over k, one considers the exponential sum S(f ) = ~Xtk)~b(f(x)). By Grothendiecks trace formula we have

The Diophantine problem for polynomial rings and fields of rational functions

We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.

Definable sets, motives and p-adic integrals

0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) of its Zp-rational points. For every n in N, there is a natural map πn : X(Zp)→ X(Z/p) assigning to a Zp-rational point its class modulo p. The image Yn,p of X(Zp) by πn is exactly the set of Z/p-rational points which can be lifted to Zp-rational points. Denote by Nn,p the cardinality of the finite set Yn,p. By a result of the first author [7], the Poincare series

Motivic Integration, Quotient Singularities and the McKay Correspondence

The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.

Hilbert’s tenth problem for quadratic rings

Let A(D) be any quadratic ring; in this paper we prove that Hilberts tenth problem for A(D) is unsolvable, and we determine which relations are diophantine over A(D).

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 also construct a canonical representative of the Milnor fibre in a suitable monodromic Grothendieck group.

Motivic exponential integrals and a Motivic Thom-Sebastiani theorem

We introduce motivic analogues of p-adic exponential integrals. We prove a basic multiplicativity property from which we deduce a motivic analogue of the Thom-Sebastiani Theorem. In particular, we obtain a new proof of the Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities of functions.

The Diophantine Problem for Polynomial Rings of Positive Characteristic

Publisher Summary This chapter discusses the diophantine problem for polynomial rings of positive characteristic. The chapter proves that the diophantine problem is unsolvable for the ring of algebraic integers in a totally real number field or in a quadratic extension of a totally real number field and shows that every recursively enumerable relation is diophantine over such a ring of algebraic integers. The diophantine problem for the field of rational functions over a formally real field is unsolvable and every recursively enumerable relation in Z[T] is diophantine over ZT]. These results are based on the fact that the diophantine problem for Z is unsolvable. The chapter examines whether the diophantine problem for the field of rationals is unsolvable.