How to turn a nonsingular cubic curve into an abelian group is not that hard to describe geometrically. The hard part, which we will not perform fully, is verifying that all of the axioms described in chapter 7 are satisfied.

By now, we have arranged our definitions so that any line intersects the curve defined by a cubic equation in 3 points, provided that we use the definition of “intersect” that we have constructed. Our goal in this chapter is to make use of some consequences of that fact.

We begin with a nonsingular cubic curve*E*, defined over a