Multiples of integral points on elliptic curves

TL;DR

This paper establishes a uniform bound on the multiples of points that can be integral on elliptic curves over integers, with implications for the distribution of integral points on twists of a fixed elliptic curve.

Contribution

It provides a bound depending only on the Tamagawa number, limiting the number of integral multiples of points on elliptic curves and analyzing integral points on twists.

Findings

Bound C depends only on Tamagawa number

At most one multiple of a point is integral beyond C

Twists of a fixed elliptic curve have at most 6 integral points in certain subgroups

Abstract

If $E$ is a minimal elliptic curve defined over $\ZZ$, we obtain a bound $C$, depending only on the global Tamagawa number of $E$, such that for any point $P\in E(\QQ)$, $nP$ is integral for at most one value of $n>C$. As a corollary, we show that if $E/\QQ$ is a fixed elliptic curve, then for all twists $E'$ of $E$ of sufficient height, and all torsion-free, rank-one subgroups $\Gamma\subseteq E'(\QQ)$, $\Gamma$ contains at most 6 integral points. Explicit computations for congruent number curves are included.