Elliptic groups and rings

TL;DR

This paper explores the algebraic structure called elliptic groups on elliptic curves, especially when classical abelian groups are insufficient, and introduces the concept of elliptic rings to deepen the understanding of these structures.

Contribution

It studies elliptic groups in detail, compares them with abelian groups, and introduces elliptic rings as a new algebraic structure related to elliptic curves.

Findings

Elliptic groups can be defined via the chord-tangent law.

When the curve has a flex point, elliptic and abelian groups coincide.

Elliptic rings are introduced as monoid objects in the category of elliptic groups.

Abstract

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 \cite{dale}, 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 so called 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 $\mathbb{Q}$), 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 at the very end even introduce the notion of an elliptic ring (a monoid object in the category of elliptic groups).