Bounds for the canonical height and integral points on elliptic curves over function fields, by Amilcar Pacheco

This paper has now appeared in Bulletin of the Australian Mathematical Society 58 (1998), 353-357 with the title `Integral points on elliptic curves over function fields of positive characteristic' and so the dvi version has been removed. In this paper we give a lower bound for the canonical height of non-torsion points of an elliptic curve over a function field of any characteristic in terms of the corresponding j-map, the genus of the function field and the discriminant of the elliptic curve. This is an analogue of a result of Hindry-Silverman ["The canonical height and integral points on elliptic curves, Invent. Math., 1988] for an elliptic curve over a function field of characteristic zero. It uses Szpiro's theorem on the discriminant of elliptic curves over function fields. We can improve their bound in the cases of the univesal elliptic curves over certain modular curves. Moreover we give a geometric condition which justifies such improvement and give examples in which it is fulfilled. Finally this result, together with a bound for the torsion subgroup of such elliptic curves and an upper bound for the height of integral points, give a bound for the number of integral points of a Weiertrass equation.

Amilcar Pacheco <amilcar@impa.br>