Elliptic Curves with Isomorphic Groups of Points over Finite Field Extensions

TL;DR

This paper characterizes the conditions under which two ordinary elliptic curves over a finite field have isomorphic groups of points over various finite field extensions, given they have the same number of points over the base field.

Contribution

It provides a complete characterization of when elliptic curves with equal base field point counts have isomorphic groups over extension fields.

Findings

Identifies specific extension degrees where groups are isomorphic

Provides criteria based on elliptic curve properties

Enhances understanding of elliptic curve group structures

Abstract

Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we characterise for which finite field extensions $\mathbb{F}_{q^k}$, $k\geq 1$ (if any) the corresponding groups of $\mathbb{F}_{q^k}$-rational points are isomorphic, i.e. $E(\mathbb{F}_{q^k}) \cong E'(\mathbb{F}_{q^k})$.