Abelian varieties of prescribed order over finite fields

Abstract

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 $#A(\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over ${\mathbb{F}}_q$. As a consequence, we generalize a result of Howe and Kedlaya for ${\mathbb{F}}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\mathbb{F}_q)$ for some abelian variety $A$ over ${\mathbb{F}}_q$. 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 $\ge q^{3\sqrt{q}\log q}$ is realizable.

References in corpus (1)

• Kernels of polarizations of abelian varieties over finite fields