Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$

TL;DR

This paper proves that every finite abelian group can be realized as the group of rational points of an ordinary abelian variety over small finite fields, with partial results over larger fields and for specific groups.

Contribution

It demonstrates that all finite abelian groups can be realized over certain small finite fields, and explores limitations and specific cases over larger fields.

Findings

Every finite abelian group occurs over $\,\mathbb{F}_2$, $\,\mathbb{F}_3$, and $\,\mathbb{F}_5$

Certain abelian groups cannot occur over large finite fields

Every finite cyclic group arises infinitely often over $\,\mathbb{F}_2$

Abstract

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\mathbb{F}_2$.