Elliptic curves over Hasse pairs

We call a pair of distinct prime powers $(q_1,q_2)=(p_1^{a_1},p_2^{a_2})$ a Hasse pair if $|\sqrt{q_1}-\sqrt{q_2}|\le 1$. For such pairs, we study the relation between the set $E_1$ of isomorphism classes of elliptic curves defined over $F_{q_1}$ with $q_2$ points, and the set $E_2$ of isomorphism classes of elliptic curves over $F_{q_2}$ with $q_1$ points. When both families $E_i$ contain only ordinary elliptic curves, we prove that their isogeny graphs are isomorphic. When supersingular curves are involved, we describe which curves might belong to these sets. We also show that if both the $q_i$'s are odd and $E_1\cup E_2\ne \varnothing$, then $E_1\cup E_2$ always contains an ordinary elliptic curve. Conversely, if $q_1$ is even, then $E_1\cup E_2$ may contain only supersingular curves precisely when $q_2$ is a given power of a Fermat or a Mersenne prime. In the case of odd Hasse pairs, we could not rule out the possibility of an empty union $E_1\cup E_2$, but we give necessary conditions for such a case to exist. In an appendix, Moree and Sofos consider how frequently Hasse pairs occur using analytic number theory, making a connection with Andrica's conjecture on the difference between consecutive primes.