Geometric decomposition of abelian varieties of order 1

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.