Ranks of quadratic twists of elliptic curves

Mark E. Watkins, Stephen Robert Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, Nicholas F. Rogers

Type: Article

Publication Date: 2015-04-20

Citations: 16

DOI: https://doi.org/10.5802/pmb.9

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.

Locations

Publications mathématiques de Besançon - View - PDF
UCL Discovery (University College London) - View - PDF
French digital mathematics library (Numdam) - View - PDF

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