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Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. 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 fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

Abstract On a projective variety defined over a global field, any Brauer–Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.