Selmer groups of quadratic twists of elliptic curves
Let \(E/\mathbb{Q}\) be an elliptic curve and \(D\) a square-free integer. \(E(D)\) will denote the elliptic curve over \(\mathbb{Q}\) which is the \(D\)-quadratic twist of \(E\) and, for an odd prime \(l\), \(S(E(D))_l\) the \(l\)-part of its Selmer group. The authors of the paper under review are focused on the frequency of the triviality of \(S(E(D))_l\). Numerical evidence suggests that \[ \#\{|D|< X\mid D\text{ square-free and }S(E(D))_l= \{1\}\}\gg_{E,l} X. \] For every elliptic curve \(E/\mathbb{Q}\) and for every prime \(l\gg_E 1\) then holds: \[ \#\{|D|< X\mid D\text{ square-free and }S(E(D))_l= \{1\}\}\gg_{E,l} \frac{\sqrt{X}} {\log X}. \] This result does not cover the case of an odd prime \(l\) for which \(E\) has rational \(l\)-torsion, i.e. \(l=3,5,7\). In this case by invoking a theorem of Frey the authors obtain analogous results if \(l=5\) or 7 and a better bound for \(l=3\).
scientific article; zbMATH DE number 1337022