Supersingular Curves With Small Non-integer Endomorphisms

Abstract

We introduce a special class of supersingular curves over ${\mathbb{F}}_{p^2}$, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved. Most notably, we show that when this set partitions into subsets in such a way that curves within each subset have small-degree isogenies between them, but curves in distinct subsets have no small-degree isogenies between them. Despite this, we show that isogenies between these curves can be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that cannot be reasonably connected by searching on $\ell$-isogeny graphs.

25 pages, 2 figures; improved bound for Theorem 1.3 (from an improvement to the proof of Proposition 4.5); new Appendix C on l-isogenies that can't be replaced by short coprime-to-l-isogenies; moved discussion of algorithms to an appendix; new, more direct (local) proofs of Lemmas 4.2 and 5.4; many minor revisions