Abstract
Given a prime power q and n ≫ 1 , we prove that every integer in a large subinterval of the Hasse-Weil interval [Formula: see text] is #A(Fq) for some ordinary geometrically simple principally polarized abelian variety A of dimension n over Fq . As a consequence, we generalize a result of Howe and Kedlaya for F2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., #A(Fq) for some abelian variety A over Fq . 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 → ∞ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if q ≤ 5 , then every positive integer is realizable, and for arbitrary q, every positive integer [Formula: see text] is realizable.