Problèmes de construction en multiplication complexe
Products of special values of modular L-functions and their applications
Properties of twists of elliptic curves.
Propriétés galoisiennes des points d'ordre fini des courbes elliptiques.
Pseudo-elliptic integrals and the values of the Weierstrass -function at torsion points.
Q-curves over quadratic fields.
Q(T)-rank of elliptic curves and certain limit coming from the local points.
Quasi-modular forms attached to elliptic curves, I
In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the...
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