Elliptic curves over function fields with a large set of integral points

TL;DR

This paper constructs elliptic curves over function fields with arbitrarily large sets of integral points, challenging existing conjectures and providing explicit examples of linearly independent points.

Contribution

It introduces new constructions of elliptic curves over function fields with large integral point sets, serving as counterexamples to a function field version of the Lang-Vojta conjecture.

Findings

Existence of elliptic curves with arbitrarily large integral point sets

Counterexamples to the function field Lang-Vojta conjecture

Explicit examples of elliptic curves with many independent points

Abstract

We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over $\mathbb_{F}_q(t)$ with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang-Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit and arbitrarily large set of linearly independent points.