Watkins, M;
Donnelly, S;
Elkies, N;
Fisher, T;
Granville, A;
Rogers, N;
(2014)
Ranks of quadratic twists of elliptic curves.
Publications mathématiques de Besançon: Algèbre et Théorie des Nombres
(2)
pp. 63-98.
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Abstract
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 a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.
| Type: | Article |
|---|---|
| Title: | Ranks of quadratic twists of elliptic curves |
| Open access status: | An open access version is available from UCL Discovery |
| Publisher version: | http://pmb.univ-fcomte.fr/2014_en.html |
| Language: | English |
| Additional information: | This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10038649 |
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