# Abelian varieties of prescribed order over finite fields

## TL;DR

This paper proves that for large enough abelian varieties over finite fields, most integers within a specific range can be realized as the number of points on these varieties.

## Contribution

The paper generalizes and improves upon prior results about realizable point counts on abelian varieties over finite fields.

## Key findings

- Every integer in a large subinterval of the Hasse–Weil interval is realizable for ordinary geometrically simple abelian varieties.
- For fixed n, the largest subinterval of realizable integers is determined asymptotically as q increases.
- Every positive integer ≥ q^{3√q log q} is realizable for arbitrary prime power q.

## Abstract

Given a prime power q and \documentclass[12pt]{minimal}
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\begin{document}$$n \gg 1$$\end{document}n≫1, we prove that every integer in a large subinterval of the Hasse–Weil interval \documentclass[12pt]{minimal}
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\begin{document}$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$\end{document}[(q-1)2n,(q+1)2n] is \documentclass[12pt]{minimal}
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\begin{document}$$\#A({\mathbb {F}}_q)$$\end{document}#A(Fq) for some ordinary geometrically simple principally polarized abelian variety A of dimension n over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_q$$\end{document}Fq. As a consequence, we generalize a result of Howe and Kedlaya for \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_2$$\end{document}F2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., \documentclass[12pt]{minimal}
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\begin{document}$$\#A({\mathbb {F}}_q)$$\end{document}#A(Fq) for some abelian variety A over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_q$$\end{document}Fq. 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 \documentclass[12pt]{minimal}
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\begin{document}$$q \rightarrow \infty $$\end{document}q→∞; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if \documentclass[12pt]{minimal}
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\begin{document}$$q \le 5$$\end{document}q≤5, then every positive integer is realizable, and for arbitrary q, every positive integer \documentclass[12pt]{minimal}
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\begin{document}$$\ge q^{3 \sqrt{q} \log q}$$\end{document}≥q3qlogq is realizable.