The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I

TL;DR

This paper investigates the probability that two 2-isogenous elliptic curves over the rationals have isomorphic groups over finite fields but not over quadratic extensions, focusing on the distribution across primes.

Contribution

It provides a probabilistic analysis of the occurrence of isomorphic groups over finite fields but not over quadratic extensions for 2-isogenous elliptic curves over rationals.

Findings

Identifies the proportion of primes where the groups are isomorphic over p but not over p^2.

Analyzes the likelihood of such occurrences under specific rationality hypotheses.

Focuses on the case =2 and quadratic extensions of finite fields.

Abstract

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over ${\mathbf{Q}}$ and asking for the proportion of primes $p$ for which $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.