Variétés rationnelles

Il s'agit d'un travail en commun avec Olivier Wittenberg.

In this talk I will report on progress on the following two questions, the first posed by Cassels in 1961 and the second considered by Bashmakov in 1974. The first question is whether the elements of the Tate-Shafarevich group are infinitely divisible when considered as elements of the Weil-Châtelet group. The second question concerns the intersection of the Tate-Shafarevich group with the maximal divisible subgroup of the Weil-Chatelet group. This is joint work with Mirela Ciperiani.

La conjecture de torsion prédit que si k est un corps de nombre etA une variété abélienne sur k alors l'ordre du sous-groupe de torsion deA(k) est borné par une constante ne dépendant que du degré de k sur Q etde la dimension de A.Cette conjecture n'est connue que pour les courbes elliptiques: Manin l'amontré en 69 pour les l-Sylow de la torsion (l:premier) puis Mazur (77),Kamienny (92), Merel (96) ont réussi a compléter la preuve en analysant lastructure des courbes modulaires X_{0}(l) (l:premier).Que les courbes elliptiques soient (essentiellement) classifiées […]

In this talk we consider the problem of counting the number of rational points of bounded height on certain intersections of two quadrics in five variables.These are del Pezzo surfaces of degree four, and we focus on the case where the surface has a conic bundle structure.

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

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 […]