Geometric decomposition of abelian varieties of order 1

TL;DR

This paper classifies and decomposes abelian varieties of order 1 over finite fields, specifically over algebraic closures of , by solving polynomial equations in roots of unity, extending prior classifications.

Contribution

It provides a detailed decomposition of abelian varieties of order 1 over algebraic closures of , building on and extending classical classification results.

Findings

Complete decomposition of abelian varieties of order 1 over algebraic closures of 

Solution of polynomial equations in roots of unity for classification

Extension of known classifications to all such varieties

Abstract

Since the 1970s, the complete classification (up to isogeny) of abelian varieties over finite fields with trivial group of rational points has been known from results of Madan--Pal and Robinson; with two exceptions these are all defined over $\mathbb{F}_2$. We determine the decomposition of these varieties into simple factors over an algebraic closure of $\mathbb{F}_2$; this requires solving a polynomial equation in three roots of unity.