On the algebra of elliptic curves

TL;DR

This paper explores the algebraic structures inherent in nonsingular elliptic curves, demonstrating that they possess a natural abelian heap structure and analyzing the associated function trusses derived from endomorphisms.

Contribution

It introduces a natural abelian heap structure on elliptic curves and links it to the algebra of endomorphisms, providing a new perspective on their intrinsic algebraic properties.

Findings

Elliptic curves admit a unique abelian heap structure.

The set of functions from an elliptic curve to itself forms a truss.

Endomorphisms of the heap determine the function algebra.

Abstract

It is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.