Super-Isolated Abelian Varieties

TL;DR

This paper characterizes super-isolated abelian varieties over finite fields using algebraic integers and proves that only finitely many such varieties exist for each dimension greater than or equal to three.

Contribution

It provides a classification of super-isolated abelian varieties via the Honda-Tate theorem and establishes finiteness results for fixed dimensions.

Findings

Finitely many super-isolated abelian varieties exist for each dimension g ≥ 3.

Characterization of these varieties using algebraic integers.

Application of Honda-Tate theorem to classify super-isolated 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 $g \geq 3$, there are finitely many such varieties.