Reed-Muller codes and supersingular curves. I
Refined Kodaira classes and conductors of twisted elliptic curves [Book]
Refined theorems of the Birch and Swinnerton-Dyer type
In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.
Reflections on Fermat's Last Theorem.
Régulateurs syntomiques et valeurs de fonctions L p-adiques II.
Régulateurs syntomiques et valeurs de fonctions Lp-adiques I.
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.
Relations between jacobians of modular curves of level
We derive a relation between induced representations on the group which implies a relation between the jacobians of certain modular curves of level . The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of .
Relative Bogomolov extensions
A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.
Relèvement selon une isogénie de système l-adiques de représentations galoisiennes associés aux motifs.
Remarks on a question of Skolem about the integer solutions of x₁x₂ - x₃x₄ = 1
Remarks on strongly modular Jacobian surfaces
In [3] we introduced the concept of strongly modular abelian variety. This note contains some remarks and examples of this kind of varieties, especially for the case of Jacobian surfaces, that complement the results of [3].
Remarques sur une conjecture de Lang
Le but de cet article est d’étudier une conjecture de Lang énoncée sur les courbes elliptiques dans un livre de Serge Lang, puis généralisée aux variétés abéliennes de dimension supérieure dans un article de Joseph Silverman. On donne un résultat asymptotique sur la hauteur des points de Heegner sur , lequel permet de déduire que la conjecture est optimale dans sa formulation.
Résolution, en nombres entiers et sous sa forme la plus générale, de l'équation cubique, homogène, à trois inconnues
Résultats récents sur la monogénéité de certains anneaux d'entiers
Rigid cohomology and -adic point counting
I discuss some algorithms for computing the zeta function of an algebraic variety over a finite field which are based upon rigid cohomology. Two distinct approaches are illustrated with a worked example.