Almost all quadratic twists of an elliptic curve have no integral points

Abstract

For a given elliptic curve $E$ in short Weierstrass form, we show that almost all quadratic twists $E_D$ have no integral points, as $D$ ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that $E$ has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.