Papers › Abelian varieties of prescribed order over finite fields
Abelian varieties of prescribed order over finite fields
Raymond van Bommel, Edgar Costa, Wanlin Li, Bjorn Poonen, Alexander Smith
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Given a prime power q and n ≫1, we prove that every integer in a large subinterval of the Hasse--Weil interval [(√(q)-1)²ⁿ,(√(q)+1)²ⁿ] is #A(𝔽_q) for some geometrically simple ordinary principally polarized abelian variety A of dimension n over 𝔽_q. 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(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) logq) is realizable.
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