  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 geometrically simple ordinary 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 $q \to \infty$; 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.
