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).

Introduction, notation

Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k-rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K. Further let (τ, B) be the K/k:-trace of J (see §2 below and).

Then the Mordell-Weil theorem (in the function field case) states that the group of K-rational points J(K) modulo the subgroup τB(k) is a finitely generated Abelian group.

Now, given Γ/K, there is a smooth projective algebraic surface S with genus g fibration f : S → C which has Γ as its generic fibre and which is relatively minimal in the sense that no fibres contain an exceptional curve of the first kind (−1-curve). It is known that the correspondence Γ/K ↔ (S, f) is bijective up to isomorphisms (cf.).

The main purpose of this paper is to give the Mordell-Weil group M = J(K)/τB(k) (modulo torsion) the structure of Euclidean lattice via intersection theory on the algebraic surface S. The resulting lattice is the Mordell- Weil lattice (MWL) of the Jacobian variety J/K, which we sometimes call MWL of the curve Γ/K or of the fibration f : S → C.

For this, we first establish the relationship between the Mordell-Weil group and the Néron-Severi group NS(S) of S (Theorem 1, stated in §2 and proved in §3). Then (in §4) we introduce the structure of lattice on the Mordell-Weil group by defining a natural pairing in terms of the intersection pairing on NS(S).