Computing heights on elliptic curves

Abstract

We describe how to compute the canonical height of points on elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over <bold>R</bold> . We describe a modified version of Tate’s series which also converges over <bold>C</bold> , and give an efficient procedure for calculating local heights at non-Archimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples.