Abelian varieties of prescribed order over finite fields
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208_2024_Article_3084.pdf
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Author(s)
van Bommel, Raymond
Costa, Edgar
Li, Wanlin
Poonen, Bjorn
Smith, Alexander
Date Issued
March 6, 2025
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Citation
van Bommel, R., Costa, E., Li, W. et al. Abelian varieties of prescribed order over finite fields. Math. Ann. 392, 1167–1202 (2025).
Version
Final published version
Abstract
Given a prime power q and n ≫ 1 , we prove that every integer in a large subinterval of the Hasse–Weil interval [ ( q - 1 ) 2 n , ( q + 1 ) 2 n ] is # A ( F q ) for some ordinary geometrically simple principally polarized abelian variety A of dimension n over F q . As a consequence, we generalize a result of Howe and Kedlaya for F 2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., # A ( F q ) for some abelian variety A over 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 → ∞ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if q ≤ 5 , then every positive integer is realizable, and for arbitrary q, every positive integer ≥ q 3 q log q is realizable.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00208-024-03084-4
