Every integer in a large subinterval of the Hasse--Weil interval is realizable

Abelian varieties of prescribed order over finite fields

Given a prime power and we prove that every integer in a large subinterval of the hasse--weil interval is for some geometrically simple ordinary principally polarized abelian variety of dimension over as a consequence, we generalize a result of howe and kedlaya for to show that for each prime power every sufficiently large positive integer is realizable, i.e., for some abelian variety 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.