Elliptic Loops

TL;DR

This paper introduces elliptic loops over local rings, extending elliptic curve addition to a broader algebraic structure, and explores their properties, stratifications, and specific cases over rings like rac{p^e}{ ext{Z}}.

Contribution

It defines elliptic loops as power associative abelian algebraic loops extending elliptic curve addition, and analyzes their structure and properties over various rings.

Findings

Elliptic loops form power associative abelian algebraic loops.

Affine parts of elliptic loops stratify into families called layers.

Structure of layers and points with same order are characterized over rings like rac{p^e}{ ext{Z}}.

Abstract

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation that extends the curve's addition. While their subset of points satisfying the curve's Weierstrass equation is a group, these larger objects are proved to be power associative abelian algebraic loops, which are seldom completely associative. When an elliptic loop has no points of order $3$, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over $R$, which we call layers. Stronger associativity properties are established when $\mathfrak{m}^e$ vanishes for small values of $e \in \mathbb{Z}$. When the underlying ring is $R = \mathbb{Z}/p^e\mathbb{Z}$, the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description.