Bounds for the integral points on elliptic curves over function fields

TL;DR

This paper establishes an upper bound on the number of integral points on elliptic curves over function fields, utilizing height bounds, algebraic rank, and sphere packing techniques, with implications from the Birch Swinnerton-Dyer conjecture.

Contribution

It introduces a new upper bound for integral points on elliptic curves over function fields, combining height bounds, algebraic rank estimates, and sphere packing optimization.

Findings

Derived an explicit upper bound in terms of conductor and field size

Extended Helfgott-Venkatesh technique to function fields

Connected algebraic and analytic ranks using Birch Swinnerton-Dyer conjecture

Abstract

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given by Silverman and extend the technique developed by Helfgott-Venkatesh to express the number of integral points on E in terms of its algebraic rank. We also use the sphere packing results to optimize the size of an implied constant. In the end we use partial Birch Swinnerton-Dyer conjecture that is known to be true over function fields to bound the algebraic rank by the analytic one and apply the explicit formula for the analytic rank of E.