Abelian varieties of prescribed order over finite fields.

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 $\#A\left(F_q\right)$ for some ordinary geometrically simple principally polarized abelian variety A of dimension n over $F_q$ . As a consequence, we generalize a result of Howe and Kedlaya for $F_2$ to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., $\#A\left(F_q\right)$ for some abelian variety A over $F_q$ . 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 $\ge q^{3\sqrt{q}logq}$ is realizable.