Integral points on quadratic twists and linear growth for certain elliptic fibrations

Abstract

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points of small naive height on quadratic twists of a fixed elliptic curve with full rational 2-torsion.