Every positive integer is the order of an ordinary abelian variety over ${\mathbb F}_2$

TL;DR

This paper proves that for every positive integer, there exists an ordinary abelian variety over the finite field with two elements that has exactly that many rational points, demonstrating a broad range of possible point counts.

Contribution

It establishes that all positive integers can be realized as the number of rational points of some ordinary abelian variety over ${f F}_2$, a previously unknown universality result.

Findings

Every positive integer is the number of rational points of some ordinary abelian variety over ${f F}_2$.

The set of possible rational point counts over ${f F}_2$ is complete for positive integers.

The result expands understanding of the distribution of rational points on abelian varieties over finite fields.

Abstract

We show that for every integer $m > 0$, there is an ordinary abelian variety over ${\mathbb F}_2$ that has exactly $m$ rational points.