Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Mark Watkins; Stephen Donnelly; Noam D. Elkies; Tom Fisher; Andrew Granville; Nicholas F. Rogers
Ranks of quadratic twists of elliptic curves
Publications mathématiques de Besançon no. 2 (2014), p. 63-98, doi: 10.5802/pmb.9
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Class. Math.: 11D25, 11D45, 11G05, 11G40, 14H52
Mots clés: courbes elliptiques, tordues quadratiques, groupes de Selmer, formule explicite, conjecture de Birch–Swinnerton-Dyer, rang algébrique

Résumé

Nous donnons un compte rendu d’un projet de grande envergure sur les rangs des courbes elliptiques dans une famille de tordues quadratiques en nous focalisant sur les courbes associées aux nombres congruents. Afin d’exclure certaines courbes, nos méthodes incluent des tests sur les 2,4,8-groupes de Selmer, l’utilisation de la formule explicite de Guinand-Weil et également des 3-descentes dans quelques cas. Nous constatons que les tordues quadratiques de rang 6 sont assez répandues (bien que toujours assez difficile à trouver), alors que celles de rang 7 semblent bien plus rares. Nous décrivons aussi notre incapacité à obtenir des tordues quadratiques de rang 8 et expliquons en quoi nos résultats peuvent se comparer à certaines prédictions sur la croissance du rang en fonction du conducteur. Enfin, nous expliquons une heuristique due à Granville, qui, lorsqu’elle est interprétée judicieusement, pourrait prédire que le rang maximal pour cette famille est en effet égal à 7.

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