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