Complex Multiplication Structure of Elliptic Curves Original Research Article

Letkbe a finite field and letEbe an elliptic curve overk. In this paper we describe, for each finite extensionlofk, the structure of the groupE(l) of points ofEoverlas a module over the ringRof endomorphisms ofEthat are defined overk. If the Frobenius endomorphismπofEoverkdoes not belong to the subringZofR, then we find thatE(l)congruent withR/R(πn−1), wherenis the degree ofloverk; and ifπdoes belong toZthenE(l) is, as anR-module, characterized byE(l)circled plusE(l)congruent withR/R(πn−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.