Elliptic curves over Hasse pairs

TL;DR

This paper investigates the relationship between elliptic curves over finite fields associated with Hasse pairs, revealing isogeny graph isomorphisms and conditions for the presence of ordinary or supersingular curves, with implications in number theory.

Contribution

It introduces the concept of Hasse pairs and analyzes the structure of elliptic curves over these pairs, including isogeny graph isomorphisms and conditions for ordinary and supersingular curves.

Findings

Isogeny graphs are isomorphic for ordinary elliptic curves over Hasse pairs.

Conditions under which supersingular curves appear in these sets.

Frequency of Hasse pairs connected to prime number conjectures.

Abstract

We call a pair of distinct prime powers $(q_1,q_2) = (p_1^{a_1},p_2^{a_2})$ a Hasse pair if $|\sqrt{q_1}-\sqrt{q_2}| \leq 1$. For such pairs, we study the relation between the set $\mathcal{E}_1$ of isomorphism classes of elliptic curves defined over $\mathbb{F}_{q_1}$ with $q_2$ points, and the set $\mathcal{E}_2$ of isomorphism classes of elliptic curves over $\mathbb{F}_{q_2}$ with $q_1$ points. When both families $\mathcal{E}_i$ 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 $\mathcal{E}_1 \cup \mathcal{E}_2 \neq \emptyset$, then $\mathcal{E}_1 \cup \mathcal{E}_2$ always contains an ordinary elliptic curve. Conversely, if $q_1$ is even, then $\mathcal{E}_1 \cup \mathcal{E}_2$ 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 $\mathcal{E}_1 \cup \mathcal{E}_2$, 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.