Mon Nov 15: 14h-15h (IHP, salle Darboux)

On s'intéresse dans ce sujet aux nouvelles méthodes de comptage qui permettent de démontrer des équivalents pour le nombre de points rationnels de hauteur bornée sur certaines variétés algébriques. On détaillera en particulier le cas de la cubique de Segre qui peut être définie comme le lieu des points satisfaisant

x₁³+x₂³+x₃³+x₄³+x₅³+x₆³=0   et   x₁ +x₂ +x₃+x₄+x₅ +x₆ =0.

Mon Sep 27: 14h-15h (IHP, salle Darboux)

On August 24-26, 2000 the Mathematics Department at U.C. Berkeley hosted a number theory conference honoring D.N. Lehmer (1867--1938), his son D.H. Lehmer (1905--1991), and his son's wife Emma Lehmer (1906--) for their contributions to mathematics and to U. C. Berkeley. Extensive information about the Lehmers was included in the conference program and the proceedings of the conference were recorded on a film which now resides in the Bancroft Library on the Berkeley campus.

In this talk I'll give a short account of the early life and education of D.N. Lehmer and some of the research activities that took place within the mathematical world at Berkeley that was created by Lehmers in the period 1900-1975, with a special focus on the theoretical and practical developments dealing with the factoring of a given integer N.

The event of most historical interest in regard to the latter was the radical shift in opinion that took place when the factoring methods deemed best by expert hand computers were put to the test on electronic computers. From these experiments the "best'' computer method was found to be one that can be called a ``generalized Fermat factoring method'', i.e., a method for finding non-trivial integers x and y for which a multiple of N is expressed as x²-y².

(Joint work with Michael A. Morrison)

Thu Sept 30: 16h30-17h30 (IHP, salle Darboux)

Consider the projective space over a finite field. A hypersurface is a variety defined by one homogenous equation having coefficients in this finite field. I will give a formula for the probability that a hypersurface of degree d is nonsingular, for d going to infinity. I will explore the proof only for curves in the projective plane, but it can easily be extended to higher dimensions.

This is a special case of a result in Bjorn Poonen's paper "Bertini Theorems over Finite Fields". But where Poonen uses the full power of algebraic geometry, this special case will be proved using only elementary facts.

Mon Nov 08: 14h-15h (IHP, salle Darboux)

Nous présenterons un résultat de modularité de certaines représentations deux-dimensionnelles du groupe de Galois absolu d'un corps de nombres totalement réel, et discuterons l'exemple fourni par une variété de Calabi-Yau tridimensionnelle non-rigide. Nous donnerons également une application à la conjecture de Bloch-Kato pour le motif adjoint associé à une forme modulaire de Hilbert.

Thu Dec 16: 14h-15h (IHP, salle Darboux)

It is well-known that the AGM can be used to evaluate elliptic integrals, hence to compute the periods of an elliptic curve. We first describe an algorithm based on Newton iterations and the AGM to evaluate modular forms in genus 1 (applications are the computation of class polynomials and of modular polynomials for example). We then show how Borchardt's generalization of the AGM to higher genus can be used to compute the Riemann matrix of a hyperelliptic curve, and -via Newton iterations- to evaluate Siegel modular forms.

Mon Nov 29: 14h-15h (IHP, salle Darboux)

We will report on the proof that there are arithmetic progression of arbitrary length in the set of primes. This is joint work with Terence Tao.

Mon Oct 25: 14h-15h (IHP, salle Darboux)

The topic of the lecture will be C_p-linear representations of global Galois groups. I will discuss Cebotarev density theorems and ramification properties for these representations and converging sequences of such representations. The results presented in the talk are joint work with Larsen and Ramakrishna, and Bellaiche, Chenevier and Larsen.

Mon Oct 18: 14h-15h (IHP, salle Darboux)

Generating functions are important to construct p-adic L-functions. Especially, at good supersingular primes for CM elliptic curves, it would be worthwhile to study the two variable generating function of Hecke L-values, since in this case the p-adic L-function of "two variable" has not yet constructed.

In this talk I will give an algebraic characterization of the two variable generating function of Hecke L-values of CM elliptic curves as a theta function corresponding to a section of the Poincaré bundle. At good ordinary primes, I will show a simple construction of the two variable p-adic L-function using this characterization. At good supersingular primes,

I will give a p-adic estimate of the denominators of the coefficients of the generating function.

Thu Dec 16: 15h-16h (IHP, salle Darboux)

Le succès des méthodes AGM pour le comptage des points sur les courbes de genre 1 et 2 en caractéristique 2 n'est plus à démontrer. Nous proposons dans cet exposé un algorithme permettant de mettre en oeuvre ce type de méthodes dans le cas des courbes de genre 3 non hyperelliptiques. Nous montrons en particulier comment on choisit un bon modèle relevant les quartiques ordinaires et comment on peut alors déterminer les thêta constantes initiales par la géométrie des bitangentes.

Fri Dec 13: 14h-15h (IHP, salle Darboux)

I will discuss a characterization of the functions of the title (classical one-variable hypergeometric series that satisfy a polynomial equation) in terms of the integrality of its Taylor coefficients. The proof follows the work of Beukers and Heckman and relies on properties of the Hermitian form fixed by the monodromy group. This Hermitian form, it turns out, can be described by an ancient construction due to Bezout.

Fri Dec 17: 14h-15h (IHP, salle Darboux)

(Joint work with Phong Nguyen)

The LLL (Lenstra-Lenstra-Lovász) algorithm has many algorithmic applications, from number theory to cryptography. Given as input d linearly independent vectors of norm less than B, the LLL algorithm computes a so-called LLL-reduced basis in polynomial time O(d⁶ log³ B), by using arithmetic operations on integers of bit-length O(d log B). This becomes rapidly too expensive in practice for lattices with even moderately large d and/or log B. For this reason, in the orthogonalisation process (central in LLL) the integer arithmetic is often replaced by floating-point arithmetic. Unfortunately, this strategy is known to be unstable in the worst-case: the correctness nor the running-time are guaranteed. In this talk we will describe a provable floating-point variant of the LLL algorithm, with running-time O(d⁵ log B (d+log B)). This is the first LLL algorithm which time grows only quadratically with respect to log B without fast integer arithmetic, like the Gaussian and Euclidean algorithms. Moreover, this algorithm is simple to describe. For these two reasons it may be seen as the "right" LLL variant.

Thu Sept 30: 14h-15h (IHP, salle Darboux)

I will discuss joint work with Harald Helfgott on upper bounds for size of the 3-torsion in the class group of a quadratic field. The key input is a method for getting improved upper bounds for the number of rational or integral points on algebraic curves.

Mon Nov 22: 14h-15h (IHP, salle Darboux)

Soit K un corps p-adique, avec p>2 et E une courbe elliptique sur K. Un théorème de Deuring affirme que la (potentielle) bonne réduction de E ne dépend que de la valuation de l'invariant modulaire absolu j(E) de E. De manière plus précise, ce resultat permet de décrire explicitement le modèle stable de la courbe.

Dans cet exposé, nous proposons une généralisation de ce résultat dans la direction suivante : on considère des paires (E,P), E étant une courbe elliptique sur K et P un point de p-torsion de E. Il est possible de définir la notion de modèle stable de (E,P), notre but étant celui de le décrire de manière explicite. Nous introduisons un invariant l attaché à (E,P) ; c'est un élément de K construit de manière analogue à l'invariant j. Le résultat central permet de décrire le modèle stable en fonction des valuations de j et de l uniquement. On en déduit, en particulier, des critères de rationalité et une nouvelle description des courbes à réduction supersingulière. On termine en donnant une interprétation modulaire de ce résultat : l'invariant l définit un revètement fini X₁(p) ---> P¹ non ramifié en dehors de trois points, qui n'est pas le revètement canonique X₁(p) ---> X(1).