On Isomorphic K-rational Groups of Isogenous Elliptic Curves over Finite Fields

TL;DR

This paper investigates conditions under which isogenous elliptic curves over finite fields have isomorphic groups of rational points, extending known results to supersingular cases and general isogenies.

Contribution

It establishes new criteria for isomorphic rational point groups among isogenous elliptic curves, including supersingular cases with specific j-invariants and a general relationship for non-equal j-invariants.

Findings

Ordinary isogenous elliptic curves with same j-invariant have isomorphic rational point groups

Supersingular elliptic curves with j-invariant 0 or 1728 also exhibit this property

A general case is proven using a result by Heuberger and Mazzoli

Abstract

We show that two ordinary isogenous elliptic curves have isomorphic groups of rational points if they have the same $j$-invariant and we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 or 1728. Using a result by Heuberger and Mazzoli we establish a general case of this relationship within isogenous elliptic curves not necessarily having equal $j$-invariant.