Abelian varieties over 𝔽 2 of prescribed order

We prove that for every positive integer m, there exist infinitely many simple abelian varieties over 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 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 resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence of suitable irreducible factors.