# Fully maximal and fully minimal abelian varieties

## TL;DR

The paper introduces a new classification of supersingular abelian varieties over finite fields into fully maximal, mixed, or fully minimal types based on Weil numbers, providing a comprehensive analysis for elliptic curves, abelian surfaces, and certain genus 3 curves.

## Contribution

It presents a novel categorization scheme for supersingular abelian varieties and thoroughly analyzes these types for specific cases including elliptic curves and abelian surfaces.

## Key findings

- Complete classification for supersingular elliptic curves.
- Analysis of supersingular abelian surfaces in arbitrary characteristic.
- Study of genus 3 supersingular curves in characteristic 2.

## Abstract

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in characteristic $2$.