ON INTEGRAL POINTS ON ISOTRIVIAL ELLIPTIC CURVES OVER FUNCTION FIELDS

Let k be a finite field and L be the function field of a curve C/k of genus g >= 1. In the first part of this note we show that the number of separable S-integral points on a constant elliptic curve E/L is bounded solely in terms of g and the size of S. In the second part we assume that L is the function field of a hyperelliptic curve C-A : s(2) = A(t), where A(t) is a square-free k-polynomial of odd degree. If infinity is the place of L associated to the point at infinity of C-A, then we prove that the set of separable {infinity}-points can be bounded solely in terms of g and does not depend on the Mordell-Weil group E(L). This is done by bounding the number of separable integral points over k(t) on elliptic curves of the form E-A : A(t)y(2) = f (x), where f (x) is a polynomial over k. Additionally, we show that, under an extra condition on A(t), the existence of a separable integral point of 'small' height on the elliptic curve E-A/k(t) determines the isomorphism class of the elliptic curve y(2) = f (x).