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

Abstract

Let E and $E^{\prime }$ be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if $E\left(F_p\right)\simeq E^{\prime }\left(F_p\right)$ but $E\left(F_{p^2}\right)\not\simeq E^{\prime }\left(F_{p^2}\right)$. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.