Seminar in Algebraic Geometry and Number Theory

Abstract: We show that almost all elliptic curves in a quadratic twist family have no integral points. Our result is conditional on a weak form of the Hall--Lang conjecture in the case of partial 2-torsion. The proof uses the reduction theory of binary quartic forms, Manin-type bounds for certain singular cubic surfaces, and character sum estimates drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve. This is joint work with Tim Browning.