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

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 ${\mathbb{F}}_2$.

accepted version

References in corpus (1)

• The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field

Cited by in corpus (1)

• Abelian varieties over finite fields and their groups of rational points