%D 2014
%P 63-98
%N 2
%T Ranks of quadratic twists of elliptic curves
%A M Watkins
%A S Donnelly
%A N Elkies
%A T Fisher
%A A Granville
%A N Rogers
%O This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
%X 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.
%J Publications mathématiques de Besançon: Algèbre et Théorie des Nombres
%L discovery10038649