Products and Polarizations of Super-Isolated Abelian Varieties

TL;DR

This paper investigates the properties of super-isolated abelian varieties over finite fields, focusing on their products and the existence and classification of principal polarizations.

Contribution

It provides characterizations of when products of super-isolated varieties are super-isolated and identifies which admit principal polarizations, including their counts.

Findings

Criteria for product super-isolation

Conditions for principal polarizations

Classification of polarized isomorphisms

Abstract

In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.