In this talk, I will explain how one can determine the asymptotic behaviour of the number of integral points on the hyperplane X_0+ ... +X_n=0 for which each coordinate is a squareful number using the classical circle method, given that n>= 4. I will also indicate how this result improves our intuition when considering the problem with only three squareful numbers.
Let C be a smooth plane cubic curve over the rationals. TheMordell--Weil Theorem can be restated as follows: there is a finitesubset B of rational points such that all rational points can beobtained from this subset by successive tangent and secantconstructions. It is conjectured that a minimal such B can bearbitrarily large
J'expliquerai un travail en commun avec Goulwen Fichou, qui consiste à mettre en place un anneau de Grothendieck K_0(BSA_R) des formules semi-algébriques grâce auquel on peut définir, sur le modèle complexe, des fonctions zêta motiviques de singularités réelles. On montre que ces fonctions zêtas sont rationnelles et que leur expression rationnelle définit des fibres de Milnor motiviques des singularités réelles. Il s'agit d'éléments de l'anneau K_0(BSA_R)otimes Z dont on montre qu'ils se réalisent, via le morphisme caractéristique d'Euler, sur la caractéristique d'Euler des fibres de Milnor ensemblistes correspondantes.
In a general set-up for non-archimedean geometry, we show how local Lipschitz continuity implies piecewise Lipschitz continuity (globally on the whole piece) for definable functions. This is joint work with G. Comte and F. Loeser which generalizes previous work by the same three authors for a fixed p-adic field in and which fits in a broader program at the interplay of arithmetic and non-archimedean geometry.
Let K/k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N_{K/k}(z). We study the Hasse principle and weak approximation for X in two cases. For =4 and P(t) irreducible over k and split in K, we prove the Hasse principle and weak approximation. For k=Q with arbitrary K, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one.
Soit K le corps de fonctions d'une courbe p-adique, G un groupe semi-simple simplement connexe sur K et X un G-torseur. Une conjecture de Colliot-Thélène, Parimala et Suresh énonce que si pour toute valuation discrète v de K, X a des points à valeurs dans le complété K_v, alors X a un K-point rationnel. Dans cet exposé, on discute cette conjecture pour les torseurs de certains groupes de types classiques. Notre méthode s'applique également au cas où K est le corps des fractions d'un anneau local intègre hensélien excellent de […]
Manjul Bhargava has recently made significant progress on the arithmetic ofelliptic curves over Q. Together with his student Arul Shankar, he has calculated the averageorder of the n-Selmer group, for n = 2,3,4,5, and has obtained an upper bound on theaverage rank (which is less than one). To do this, they identify elements of the Selmer groupwith certain orbits in a representation of a semi-simple group over Q, and estimatethe number of orbits of bounded height using the geometry of numbers. In this talk, which is a report on joint […]
voir sur les archives du séminaire.
