2021/06/25 by Raymond van Bommel, van Bommel, Raymond, Edgar Costa +7
Computer Science · Mathematics · #11Y99 #14G15 #14K15 #31A15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary 11G10 #Secondary 11G25
paper · pdf · doi:10.48550/arxiv.2106.13651
openalex publication_date 2021/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\mathbbFq) for some geometrically simple ordinary principally polarized abelian variety A of dimension n over \mathbbFq. As a consequence, we generalize a result of Howe and Kedlaya for \mathbbF2 to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., #A(\mathbbFq) for some abelian variety A over \mathbbFq. 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 ≥ q3 √(q) log q is realizable.