Density of rational points on elliptic surfaces

TL;DR

This paper proves that for certain elliptic surfaces over number fields, rational points are dense outside a computably small subset, extending previous results and providing a universal version over twists and extensions.

Contribution

It establishes a new density result for rational points on elliptic surfaces with two fibrations, generalizing prior work by Swinnerton-Dyer and others.

Findings

Existence of a computable proper subset Z of V for density of rational points

Density of rational points outside Z for all extensions of degree up to d

A universal version of the density statement over twists and extensions

Abstract

Suppose V is a surface over a number field k that admits two elliptic fibrations. We show that for each integer d there exists an explicitly computable closed subset Z of V, not equal to V, such that for each field extension K of k of degree at most d over the field of rational numbers, the set V(K) is Zariski dense as soon as it contains any point outside Z. We also present a version of this statement that is universal over certain twists of V and over all extensions of k. This generalizes a result of Swinnerton-Dyer, as well as previous work of Logan, McKinnon, and the author.