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

TL;DR

This paper demonstrates that for any positive integer, there are infinitely many simple abelian varieties over the finite field  with that order, using a constructive method based on Weil polynomials and binary representations.

Contribution

It provides a constructive proof showing the existence of infinitely many simple abelian varieties over  of any prescribed order, extending previous results for the case m=1.

Findings

Existence of infinitely many simple abelian varieties over  for any order m.

Explicit construction of Weil polynomials corresponding to these varieties.

Use of binary representations to control polynomial properties and ensure irreducibility.

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.