Calculating Root Numbers of Elliptic curves over Q.
Class fields of abelian extensions of Q.
Class invariants of Mordell-Weil groups.
Class numbers of quadratic fields and Shimura's correspondence.
CM liftings of supersingular elliptic curves
Assuming GRH, we present an algorithm which inputs a prime and outputs the set of fundamental discriminants such that the reduction map modulo a prime above from elliptic curves with CM by to supersingular elliptic curves in characteristic is surjective. In the algorithm we first determine an explicit constant so that implies that the map is necessarily surjective and then we compute explicitly the cases .
Compatible families of elliptic type
Completely faithful Selmer groups over Kummer extensions.
Computation of the Selmer groups of certain parametrized elliptic curves
Computer-aided serendipity
Computing integral points on elliptic curves
Computing integral points on Mordell's elliptic curves.
Computing modular degrees using -functions
We give an algorithm to compute the modular degree of an elliptic curve defined over . Our method is based on the computation of the special value at of the symmetric square of the -function attached to the elliptic curve. This method is quite efficient and easy to implement.
Computing periods of cusp forms and modular elliptic curves.
Computing the modular degree of an elliptic curve.
Congruent numbers over real number fields
It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.
Constructions de polynômes génériques à groupe de Galois résoluble
On sait que les seuls sous-groupes résolubles transitifs du groupe symétrique ₅ sont isomorphes au groupe de Frobenius , au groupe diédral D₅ et au groupe cyclique C₅. Nous montrerons comment construire des extensions de degré 5 à groupe de Galois résoluble à l’aide de courbes elliptiques. Dans un premier paragraphe nous utiliserons une courbe elliptique ayant un point de 5-torsion rationnel pour les groupes D₅ et C₅. Puis, dans le paragraphe suivant, nous utiliserons une courbe elliptique ayant...
Corps engendré par les points de 13-torsion des courbes elliptiques
Correspondance
Corrigendum to "Distribution of the traces of Frobenius on elliptic curves over function fields" (Acta Arith. 106 (2003), 255-263)
Corrigendum to the paper "The number of solutions of the Mordell equation" (Acta Arith. 88 (1999), 173–179)