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**Skills and Expertise**

## Publications

Publications (88)

We give an algebraic geometric proof of the Theorem of Ax and Kochen on
p-adic diophantine equations in many variables. Unlike Ax-Kochen's proof, ours
does not use any notions from mathematical logic and is based on weak
toroidalization of morphisms. We also show how this geometric approach yields
new proofs of the Ax-Kochen-Ersov transfer principl...

We give a short proof of Macintyre's Theorem on Quantifier Elimination for the p-adic numbers, using a version of monomialization that follows directly from the Weak Toroidalization Theorem of Abramovich an Karu (extended to non-closed fields).

This note contains some results related to the definitions of toroidal
embeddings and toroidal morphisms over non-closed fields of
characteristic zero.

We give a short proof of Macintyre's Theorem on Quantifier Elimination for
the p-adic numbers, using a version of monomialization that follows directly
from the Weak Toroidalization Theorem of Abramovich an Karu (extended to
non-closed fields).

We prove a conjecture of Colliot-Th\'el\`ene that implies the Ax-Kochen
Theorem on p-adic forms. We obtain it as an easy consequence of a diophantine
purity theorem whose proof forms the body of the present paper.

We prove that any dominant morphism of algebraic varieties over a field k of
characteristic zero can be transformed into a toroidal (hence monomial)
morphism by projective birational modifications of source and target. This was
previously proved by the first and third author when k is algebraically closed.
Moreover we show that certain additional r...

For a linear group $G$ acting on an absolutely irreducible variety $X$ over the rationals $\QQ$, we describe the orbits of $X(\QQ_p)$ under $G(\QQ_p)$ and of $X(\FF_p((t)))$ under $G(\FF_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\QQ_p$ or $\FF_p((t))$ is $\leq 0$ for $p$ big enoug...

We present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology.
The paper vastly generalizes previous work since in practice all known cases, for example, hyperelliptic, superelliptic, and
Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed...

We present an algorithm to compute the zeta function of an arbitrary
hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. Given a genus g hyperelliptic curve
defined over Fqn, the average-case time complexity is O(g4 + ε n3 + ε) and the average-case space complexity is O...

We describe an algorithm to compute the zeta function of any Cab curve over any finite field Fpn. The algorithm computes a p-adic approximation of the characteristic polynomial of Frobenius by computing in the Monsky–Washnitzer cohomology of the curve and thus generalizes Kedlaya's algorithm for hyperelliptic curves. For fixed p the asymptotic runn...

We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton polyhedron. In particular, up to an easy non-zero factor, this residue only depends on the (tau_0)-principal part o...

This thesis focuses on p-adic algorithms to compute zeta functions of curves over nite elds. The zeta function contains important arithmetic and geometric information about the curve and its Jacobian. Motivated by practical applications such as cryptography and coding theory, ecient algorithms to compute the zeta function have become increasingly i...

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 unders...

this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R

this article we investigate their asymptotic behaviour when the parameter t tends to infinity in terms of the geometry of the Newton polyhedron of the phase f . It is well-known that the greatest contributions to this asymptotic behaviour arise from the critical points of f : if f is regular on the support of # subsequent oscillations will more or...

We illustrate the principle: rational generating series occuring in arithmetic geometry are motivic in nature.

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 type depend on p.

In this paper we present an extension of Kedlaya’s algorithm for computing the zeta function of an Artin-Schreier curve over a finite field \(
\mathbb{F}_q
\)
of characteristic 2. The algorithm has running time O(g
5+ɛ log3+ɛq) and needs O(g
3 log3q) storage space for a genus g curve. Our first implementation in MAGMA shows that one can now generat...

Introduction Let X be an algebraic variety, not necessarily smooth, over a eld k of characteristic zero. We denote by L(X) the k-scheme of formal arcs on X : K-points of L(X) correspond to formal arcs Spec K[[t]] ! X , for K any eld containing k. In a recent paper [8], we developped an integration theory on the space L(X) with values in c M, a cert...

We prove estimates on exponential sums modulo p(n), associated to polynomials that are nondegenerate with respect to their Newton polyhedron. Our bounds are uniform in p and n, and yield special cases of Igusa's conjecture on exponential sums.

Introduction Let X be an algebraic variety, not necessarily smooth, over a eld k of characteristic zero. We denote by L(X) the k-scheme of formal arcs on X : K-points of L(X) correspond to formal arcs Spec K[[t]] ! X , for K any eld containing k. In a recent paper [8], we developped an integration theory on the space L(X) with values in c M, a cert...

Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is isomorphic to S 0 , [S] = [S n S 0 ] + [S 0 ] if S 0 is c...

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.

Introduction 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(Z p ) of its Z p -rational points. For every n in N, there is a natural map n : X(Z p ) ! X(Z=p n+1 ) assigning to a Z p -rational point its class modulo p n+1 . The image Y n;p of X(Z p ) by n is exactly the set of Z=...

We give a very explicit formula for Igusa's local zeta function Zf(s) associated to a polynomial f in several variables over the p-adic numbers, when f is sufficiently non-degenerated with respect to its Newton polyhedron Γ(f). Using this formula and the estimates of Adolphson and Sperber on exponential sums over finite fields, we study the largest...

We prove several results on monodromies associated to Macdonald integrals, that were used in our previous work on the finite field analogue of a conjecture of Macdonald. We also give a new proof of our formula expressing recursively the zeta function of the local monodromy at the origin of the discriminant of a finite Coxeter group in terms of the...

We show how Resolution of Singularities in characteristic p implies the decidability of the existential theory of Fp [[t]] in the language of discrete valuation rings, where t is a single variable and Fp the p- element field.

To a polynomial f over a p--adic field K and a character of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s; f; ), which is a meromorphic function on C . Several theorems and conjectures relate the poles of Z(s; f; ) to the monodromy of f ; the so--called holomorphy conjecture states roughly that if the...

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.

We survey applications of quantifier elimination to number theory and algebraic geometry, focusing on results of the last 15 years. We start with the applications of p-adic quantifier elimination to p-adic integration and the rationality of several Poincar series related to congruences f(x) = 0 modulo a prime power, where f is a polynomial in sever...

We associate canonical virtual motives to definable sets over a field of characteristic zero. We use this construction to show that very general p-adic integrals are canonically interpolated by motivic ones.

this paper. To achieve this aim, we enlarge slightly our virtualmotives by attaching virtual motives to Gauss sums in a way very similar to Anderson'sconstruction of ulterior motives [1]. But now equation (1.1) is no longer trivial, andthe main result of the paper is that it still holds true for our motivic exponentialintegrals (Theorem 4.2.4). We...

Let G be a complex linear algebraic group and ρ:G [rightward arrow] GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character φ [set membership] Hom (G, C$^×$), meaning that f ^ ρ (g) =φ (g) f for all g in G. Choose a no...

In this Note we obtain some results and state a conjecture on the vanishing of principal value integrals over local fields. These integrals were first introduced by Langlands. They appear as the coefficients of the asymptotic expansion of fibre integrals and of oscillating integrals. They also appear as residues of poles of Igusa's local zeta funct...

this paper. To achieve this aim, we enlarge slightly our virtual motives by attaching virtual motives to Gauss sums in a way very similar to Anderson's construction of ulterior motives [1]. But now equation (1.1) is no longer trivial, and the main result of the paper is that it still holds true for our motivic exponential integrals (Theorem 4.2.4)....

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 the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with mot...

We prove a character sum identity for Coxeter arrangements which is a finite field analogue of Macdonald's conjecture proved by Opdam.

To a polynomial f over a p-adic field K and a character χ of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s, f, χ), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s, f, χ) to the monodromy of f; the so-called holomorphy conjecture states roughly that if the...

In this paper we shall consider some decision problems for ordinary differential equations. All differential equations will be algebraic differential equations, i.e. equations of the form P(x, y, y′, …, y ⁽ⁿ⁾ ) = 0 (or P ( x , y 1 , …, y m , y ′ 1 …, y ′ m , …) = 0 in the case of several dependent variables), where P is a polynomial in all its vari...

We extend some results of Christol and Furstenberg to the case of several variables: (1) A power series in several variables over the p-adic integers p is congruent mod ps to an algebraic power series if and only if its coefficients satisfy certain congruences mod ps. (2) Any algebraic power series in m variables over a field K can be written as th...

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...

Let $k$ be a field of characteristic $0, k\lbrack\lbrack X_1, X_2\rbrack\rbrack$ the ring of formal power series and $R = k\lbrack\lbrack X_1, X_2\rbrack\rbrack\lbrack X_3, X_4, X_5\rbrack$ the algebraic closure of $k\lbrack\lbrack X_1, X_2\rbrack\rbrack\lbrack X_3, X_4, X_5\rbrack$ in $k\lbrack\lbrack X_1,\ldots,X_5\rbrack\rbrack$. It is shown tha...

This paper studies the problem of smoothing a homomorphis.n of commutative rings along a section. The data needed to pose the problem make up a commutative diagram of al:fine schemes, such that Y is finitely presented over X. Our standard notation is that X, X, Y are the spectra of A, Ā, B respectively, and that B is a finitely presented A-algebra....

We prove that $Z$ is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.

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.

Let Z[T] be the ring of polynomials with integer coefficients. We prove that every recursively enumerable subset of Z[T] is diophantine over Z[T]. This extends a theorem of Davis and Putnam which states that every recursively enumerable subset of Z is diophantine over 1[T].

Let Z [ T ] {\mathbf {Z}}[T] be the ring of polynomials with integer coefficients. We prove that every recursively enumerable subset of Z [ T ] {\mathbf {Z}}[T] is diophantine over Z [ T ] {\mathbf {Z}}[T] . This extends a theorem of Davis and Putnam which states that every recursively enumerable subset of Z is diophantine over Z [ T ] {\mathbf {Z}...

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

vating the abstract developments. Applied model theory is using ideas and methods from other parts of mathematics, ranging from homology theory to complex analytic geometry. These two strands of research were exhibited at the BIRS workshop. The workshop was used as an opportunity to exhibit and elucidate two large pieces of technical work which hav...

"Workshop on Hilbert's tenth problem : relations with arithmetic and algebraic geometry, November 2-5, 1999, Ghent Uni versity, Belgium" Incluye bibliografía