Complex Multiplication Structure of Elliptic Curves

Let k be a finite field and let E be an elliptic curve over k. In this paper we describe, for each finite extension l of k, the structure of the group E(l) of points of E over l as a module over the ring R of endomorphisms of E that are defined over k. If the Frobenius endomorphism ? of E over k does not belong to the subring Z of R, then we find that E(l)$R R(?&1), where n is the degree of l over k; and if ? does belong to Z then E(l) is, as an R-module, characterized by E(l) E(l )$R R(?&1). The arguments used in the proof of these statements generalize to yield a description of the group of points of an elliptic curve over an algebraically closed field as a module over suitable subrings of the endomorphism ring of the curve. It is shown that straightforward generalizations of the results of this paper to abelian varieties of dimension greater than 1 cannot be expected to exist. 1996 Academic Press, Inc.