# Abelian varieties of prescribed order over finite fields

@inproceedings{Bommel2021AbelianVO, title={Abelian varieties of prescribed order over finite fields}, author={Raymond van Bommel and Edgar Costa and Wanlin Li and Bjorn Poonen and Alexander D. Smith}, year={2021} }

Given a prime power q and n ≫ 1, we prove that every integer in a large subinterval of the Hasse–Weil interval [( √ q − 1), (√q + 1)] is #A(Fq) for some geometrically simple ordinary principally polarized abelian variety A of dimension n over Fq. As a consequence, we generalize a result of Howe and Kedlaya for F2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., #A(Fq) for some abelian variety A over Fq. Our result also improves upon the best… Expand

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