Products of special values of modular L-functions and their applications
Properties of twists of elliptic curves.
Rank zero quadratic twists of modular elliptic curves
Ranks of quadratic twists of an elliptic curve
Ranks of quadratic twists of elliptic curves
We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...
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.
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.
Selmer groups and Heegner points in anticyclotomic -extensions
Some analogies between number theory and dynamical systems on foliated spaces.
Studying the growth of Mordell-Weil.
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 les mauvais facteurs locaux des fonctions L attachées aux surfaces abéliennes et surfaces K-3
Symmetric square -functions and Shafarevich-Tate groups.
Symmetric squares of elliptic curves: rational points and Steiner groups.
Systèmes d’Euler -adiques et théorie d’Iwasawa
On définit la notion de système d’Euler associé à une représentation -adique du groupe de Galois absolu de dans le cas cyclotomique. Cette notion a été introduite par Kolyvagin. L’existence d’un tel système a des conséquences très importantes sur l’étude des groupes de Selmer de que nous développons ici.
Tamagawa numbers for motives with (non-commutative) coefficients.
Tate-Shafarevich groups of the congruent number elliptic curves
The Analytic Rank of a Family of Jacobians of Fermat Curves
We study the family of curves , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves . As a corollary we conclude that the jacobians of the curves with even analytic rank and those with odd analytic rank are equally distributed.