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

Abstract

Let $\ell$ be a prime number and let $E$ and $E^{\prime }$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p\ne \ell$. Suppose the groups $E(k)$ and $E^{\prime }(k)$ are isomorphic, but $E(K)\not\simeq E^{\prime }(K)$, 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 $E$, $E^{\prime }$ over $\mathbf{Q}$ and asking for the proportion of primes $p$ for which $E({\mathbf{F}}_p)\simeq E^{\prime }({\mathbf{F}}_p)$ and $E({\mathbf{F}}_{p^2})\not\simeq E^{\prime }({\mathbf{F}}_{p^2})$.