Formal groups, elliptic curves, and some theorems of Couveignes

Abstract

The formal group law of an elliptic curve has seen recent applications to computational algebraic geometry in the work of Couveignes to compute the order of an elliptic curve over finite fields of small characteristic ([2], [6]). The purpose of this paper is to explain in an elementary way how to associate a formal group law to an elliptic curve and to expand on some theorems of Couveignes. In addition, the paper serves as background for [1]. We treat curves defined over arbitrary fields, including fields of characteristic two or three. The author wishes to thank Al Laing for a careful reading of an earlier version of the manuscript and for many useful suggestions.