Abelian varieties of prescribed order over finite fields

Given a prime power q and $$ , we prove that every integer in a large subinterval of the Hasse-Weil interval $$ is $$ for some ordinary geometrically simple principally polarized abelian variety A of dimension n over $$ . As a consequence, we generalize a result of Howe and Kedlaya for $$ to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., $$ for some abelian variety A over $$ . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse-Weil interval. A separate argument determines, for fixed n, the largest subinterval of the Hasse-Weil interval consisting of realizable integers, asymptotically as $$ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $$ , then every positive integer is realizable, and for arbitrary q, every positive integer $$ is realizable.