Rang de courbes elliptiques avec groupe de torsion non trivial
On construit des courbes elliptiques sur de rang au moins 3, avec un sous-groupe de torsion non trivial. Par spécialisation, des courbes elliptiques de rang 5 et 6 sur sont obtenues.
Rang des courbes elliptiques et sommes d'exponentielles.
Rang moyen de familles de courbes elliptiques et lois de Sato-Tate.
Rank computations for the congruent number elliptic curves.
Rank frequencies for quadratic twists of elliptic curves.
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 elliptic curves in cubic extensions
Ranks of elliptic curves in families of quadratic twists.
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...
Rational elliptic curves are modular
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.