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

The paper advances the realization problem from merely the order of rational points to the full group structure, proving that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over the small fields $\mathbb{F}_2$, $\mathbb{F}_3$, and $\mathbb{F}_5$. It develops and exploits square-free isogeny-class techniques together with endomorphism-algebra descriptions to construct abelian varieties with prescribed point-structure, and extends these ideas to broader finite fields where possible. Key contributions include a complete realization for the small fields, explicit bounds and mechanisms in the general $\mathbb{F}_q$ setting, and infinite families of simple varieties over $\mathbb{F}_2$ realizing any cyclic group as a rational-points group. The results deepen the understanding of how Frobenius, Weil polynomials, and endomorphism rings govern the possible groups of rational points on abelian varieties over finite fields, with implications for isogeny classes and explicit constructions.

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 $\mathbb{F}_2$.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Proposition 2.5
• Definition 2.6
• Proposition 2.7
• proof
• Remark 2.8