Rang moyen de familles de courbes elliptiques et lois de Sato-Tate.
Rank of elliptic curves associated to Brahmagupta quadrilaterals
We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.
Rank zero quadratic twists of modular elliptic curves
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...
Rational Isogenies of Prime Degree.
Rational points and Dirichlet series. (Points rationnels et séries de Dirichlet.)
Rational points on elliptic curves over Q in elementary abelian 2-extensions of Q.
Rational points on .
Rational torsion points on elliptic curves over number fields
Reductions of an elliptic curve with almost prime orders
Regularity of the four dimensional Sklyanin algebra
Résolution, en nombres entiers et sous sa forme la plus générale, de l'équation cubique, homogène, à trois inconnues
Selmer groups and Heegner points in anticyclotomic -extensions
Selmer groups and torsion zero cycles on the selfproduct of a semistable elliptic curve.
Séries d'Eisenstein et fonctions L de courbes elliptiques à multiplication complexe.
Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas
This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.
Solving the equation x3 - y2 = r in number fields.
Some estimates from étale cohomology.
Some functions related to the derivatives of the L-series of an elliptic curve at s=1
Stickelberger elements and modular parametrizations of elliptic curves.