Abelian varieties over $\mathbb{F}_2$ of prescribed order

Abstract

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit sequence of Weil polynomials giving rise to abelian varieties over $\mathbb{F}_2$ of order $m$. This sequence itself depends on the choice of a suitable generalized binary representation of $m$; by making careful choices of this representation, we can ensure that the the resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence of suitable irreducible factors.