Super-Isolated Abelian Varieties

Abstract

We call an abelian variety over a finite field ${\mathbb{F}}_q$ super-isolated if its (${\mathbb{F}}_q$-rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension $g\ge 3$, there are finitely many such varieties.

Fixed various typos, added additional motivation to the introduction, and rewrote parts of section 5 to be more clear