Abstract

We study the integral points of an elliptic curve over function fields from the viewpoint of Mordell–Weil lattices. On the one hand, it leads to a surprisingly simple determination of all integral points in some favorable situation. On the other hand, it gives a method to produce elliptic curves with ‘many’ integral points.

Introduction

The finiteness of the set of integral points of an elliptic curve, defined by a Weierstrass equation with integral coefficients in a number field, is due to Siegel; an effective bound was given by Baker.

The function field analogue of this fact is known. It is indeed considerably easier to prove, with stronger effectivity results. See Hindry & Silverman (1988), Lang (1990), Mason (1983) for example.

Yet it will require in general some nontrivial effort to determine all the integral points (e.g. with polynomial coordinates) of a given elliptic curve over a function field.

The purpose of this paper is to study this question from the viewpoint of Mordell–Weil lattices. Sometimes it gives a very simple determination of integral points. For example, we can show that the elliptic curve

defined over K = C(t) has exactly 240 ‘integral points’ P = (x, y) such that x, y are polynomials in t, and they are all of the form

In fact, it has been known for some time that the structure of the Mordell–weil lattice in question on E(K) is isomorphic to the root lattice E8 of rank 8 (see e.g. Shioda 1991a) and the rational points corresponding to the 240 roots of E8 are integral points of the above form (see Lemma 10.5, Lemma 10.9 and Theorem 10.6 in Shioda 1990).