The difference between the Weil height and the canonical height on elliptic curves

Abstract

Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the <italic>j</italic> -invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples.