Rational points and Dirichlet series. (Points rationnels et séries de Dirichlet.)
Rational Points of Finite Order on Elliptic Curves.
Rational points on certain elliptic surfaces
Rational points on some elliptic surfaces
Rational points on .
Rational points on
Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of , for and a prime number exceeding . This includes the case of the curves . We then prove, with the help of computer calculations, that the same holds true for in the range , . The combination of those results completes the qualitative study of rational points on undertook in our previous work, with the only exception of .
Rational torsion points on elliptic curves over number fields
Rational triangles with equal area.
Ratios of congruent numbers
Reductions of an elliptic curve with almost prime orders
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.
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.
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
-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger
In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.
-integral solutions to a Weierstrass equation
The rational solutions with as denominators powers of to the elliptic diophantine equation are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term (-) unit equations with special properties, that are solved by linear forms in real and -adic logarithms.
Selmer groups and coupling; particular case of elliptic curves. (Groupes de Selmer et accouplements; cas particulier des courbes elliptiques.)
Selmer groups and Heegner points in anticyclotomic -extensions