Given a prime power q and \(n \gg 1\) , we prove that every integer in a large subinterval of the Hasse–Weil interval \([(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]\) is \(\#A({\mathbb {F}}_q)\) for some ordinary geometrically simple principally polarized abelian variety A of dimension n over \({\mathbb {F}}_q\) . As a consequence, we generalize a result of Howe and Kedlaya for \({\mathbb {F}}_2\) to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., \(\#A({\mathbb {F}}_q)\) for some abelian variety A over \({\mathbb {F}}_q\) . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixed n, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as \(q \rightarrow \infty \) ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if \(q \le 5\) , then every positive integer is realizable, and for arbitrary q, every positive integer \(\ge q^{3 \sqrt{q} \log q}\) is realizable.
