On the proportion of rank 0 twists of elliptic curves

Dąbrowski, Andrzej

doi:10.1016/j.crma.2008.03.025

Number Theory

On the proportion of rank 0 twists of elliptic curves
[Sur la proportion de tordues de courbes elliptiques qui sont de rang 0]

Présenté par : Jean-Pierre Serre

Dąbrowski, Andrzej ¹

¹ Institute of Mathematics, University of Szczecin, ul. Wielkopolska 15, 70-451 Szczecin, Poland

Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 483-486.

Résumé
Abstract

Soit E une courbe elliptique définie sur Q, $E_{d}$ la tordue quadratique de E par d, et $r_{E_{d}} : = rang E_{d} (Q)$ . On démontre qu'il existe, pour tout entier positif k, des courbes elliptiques $E^{1}, \dots, E^{k}$ , qui sont 2 à 2 non isogènes, et telles que $r_{{E^{1}}_{p}} = \dots = r_{{E^{k}}_{p}} = 0$ pour une famille de nombres premiers p de densité positive.

Let E be an elliptic curve defined over Q, let $E_{d}$ denote its dth quadratic twist, and $r_{E_{d}} : = rank E_{d} (Q)$ . We prove, that, for any positive integer k there are pairwise non-isogenous elliptic curves $E^{1}, \dots, E^{k}$ such that $r_{{E^{1}}_{p}} = \dots = r_{{E^{k}}_{p}} = 0$ for a positive proportion of primes p.

Reçu le : 2006-11-07
Accepté le : 2008-03-25
Publié le : 2008-04-17

DOI : 10.1016/j.crma.2008.03.025

Affiliations des auteurs :

Dąbrowski, Andrzej ¹

¹ Institute of Mathematics, University of Szczecin, ul. Wielkopolska 15, 70-451 Szczecin, Poland

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Dąbrowski, Andrzej. On the proportion of rank 0 twists of elliptic curves. Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 483-486. doi : 10.1016/j.crma.2008.03.025. http://www.numdam.org/articles/10.1016/j.crma.2008.03.025/

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