Abelian varieties of prescribed order over finite fields

Abstract Given a prime power q and $$n \gg 1$$ n ≫ 1 , we prove that every integer in a large subinterval of the Hasse–Weil interval $$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ [ ( q - 1 ) 2 n , ( q + 1 ) 2 n ] is $$\#A({\mathbb {F}}_q)$$ # A ( F q ) for some ordinary geometrically simple principally polarized abelian variety A of dimension n over $${\mathbb {F}}_q$$ F q . As a consequence, we generalize a result of Howe and Kedlaya for $${\mathbb {F}}_2$$ F 2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., $$\#A({\mathbb {F}}_q)$$ # A ( F q ) for some abelian variety A over $${\mathbb {F}}_q$$ 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 $$q \rightarrow \infty $$ q → ∞ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $$q \le 5$$ q ≤ 5 , then every positive integer is realizable, and for arbitrary q, every positive integer $$\ge q^{3 \sqrt{q} \log q}$$ ≥ q 3 q log q is realizable.

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