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

TL;DR

This paper proves linear growth in the number of rational points on specific del Pezzo surfaces, confirming Manin's conjecture, and studies integral points on quadratic twists of elliptic curves with full rational 2-torsion.

Contribution

It establishes linear growth of rational points on certain del Pezzo surfaces and analyzes integral points on quadratic twists of elliptic curves, providing new insights into these areas.

Findings

Number of rational points grows linearly with height

Average integral points on quadratic twists are bounded

Supports Manin's conjecture for specific surfaces

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.