Elliptic groups and rings

As it is well known, one can define an abelian group on the points of an elliptic curve, using the so called chord–tangent law (Husemöller in Elliptic curves, 2002), and a chosen point. However, that very chord–tangent law allows us to define a rather more obscure algebraic structure, which we call an elliptic group, on the points of an elliptic curve. In the cases when our curve has a flex point (intersection number with the tangent is 3), the classical abelian group and the elliptic group carry the same information. However, if our curve does not have such a point (which often happens over \(\mathbb {Q}\) $\mathbb{Q}$ ), the abelian group is not enough to recover the elliptic group. The aim of this paper is to study this algebraic structure in more detail, its connections to abelian groups and even introduce the notion of an elliptic ring (a monoid object in the category of elliptic groups).