Supersingular primes for elliptic curves over real number fields
Sur la complexité du principe de Tarski-Seidenberg
Sur la complexité du principe de Tarski-Seidenberg
Sur le nombre des points rationnels de hauteur borné des variétés algébriques.
Sur le principe de Hasse et l'approximation faible, et sur une hypothèse de Schinzel
Sur le rang des jacobiennes sur un corps de fonctions
Soit une surface projective fibrée au-dessus d’une courbe et définie sur un corps de nombres . Nous donnons une interprétation du rang du groupe de Mordell-Weil sur de la jacobienne de la fibre générique (modulo la partie constante) en termes de moyenne des traces de Frobenius sur les fibres de . L’énoncé fournit une réinterprétation de la conjecture de Tate pour la surface et généralise des résultats de Nagao, Rosen-Silverman et Wazir.
Sur le théorème de Runge
Terms in elliptic divisibility sequences divisible by their indices
The Brauer group and the Brauer–Manin set of products of varieties
Let and be smooth and projective varieties over a field finitely generated over , and let and be the varieties over an algebraic closure of obtained from and , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br Br( has finite index in the Galois invariant subgroup of Br. This implies that the cokernel of the natural map Br Br Br is finite when is a number field. In this case we prove that the Brauer–Manin set of the product of...
The Brauer-Manin obstruction on a general diagonal quartic surface
The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer.
The canonical height and integral points on elliptic curves.
The defect of weak approximation for homogeneous spaces
The density of rational and integral points on algebraic varieties
The Eisenstein descent on J0 (N).
The equation xyz = x + y + z = 1 in integers of a cubic field.
The existence of infinitely many supersingular primes for every elliptic curve over Q.
The Hasse problem for rational surfaces.
The Hilbert Modular Surface for the Ideal (...5) and the Horrocks-Mumford Bundle.
The -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
In this paper we show that for every prime the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an elliptic curve defined over a number field , with bounded by a constant depending only on . From this we deduce that the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an abelian variety, with bounded by a constant depending only on .