Every finite abelian group is the group of rational points of an ordinary abelian variety over F2, F3 and F5

Abstract

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F 2, F 3 and F 5. We produce partial results for abelian varieties over a general finite field F q. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over F q when q is large.