Tim Browning: 100% of quadratic twists have no integral points

Speaker
Tim Browning, IST Austria

Abstract
In recent joint work with Stephanie Chan, for a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points. 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 the reduction theory of binary quartic forms, Manin-type bounds for certain singular cubic surfaces, and Heath-Brown’s character sum yoga (as used in his analysis of Selmer groups for the congruent number curve).