Abelian varieties of prescribed order over finite fields

Given a prime power q and n≫1, we prove that every integer in a large subinterval of the Hasse–Weil interval [√q−1)2n,(√q+1)2n] 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 ≥ q3√q log q is realizable.