The probability of isomorphic group structures of isogenous elliptic curves over finite fields

TL;DR

This paper calculates the likelihood that two isogenous elliptic curves over finite fields have isomorphic group structures over those fields, using advanced algebraic techniques.

Contribution

It provides a precise determination of the proportion of primes where the groups of points of two isogenous elliptic curves are isomorphic.

Findings

Explicit formula for the proportion of primes with isomorphic groups

Extension of previous techniques to new elliptic curve scenarios

Enhanced understanding of elliptic curve group structures over finite fields

Abstract

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in \cite{ck} and \cite{rnt}.