2019/07/31 by Jungin Lee
Computer Science · Mathematics · #Abelian group #Abelian variety #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #Dimension (graph theory) #Finite field #Isogeny #Logarithm #Simple (philosophy) #math.NT #msc:11G10 #msc:11G25 #msc:14K02
paper · pdf · doi:10.1007/s00209-020-02476-x
published as Mathematische Zeitschrift 296 (2020), 685-693 · 9 pages, to appear in Math. Z
openalex created_date 2019/07/23 · arxiv created 2019/11/06 · openalex publication_date 2020/02/19 · arxiv updated 2020/09/01 · openalex updated_date 2026/08/05
In this paper, we investigate the asymptotic behavior of the number sq(g) of isogeny classes of simple abelian varieties of dimension g over a finite field \mathbbFq. We prove that the logarithmic asymptotic of sq(g) is the same as the logarithmic asymptotic of the number mq(g) of isogeny classes of all abelian varieties of dimension g over \mathbbFq. We also prove that \limsupg → ∞ (sq(g))/(mq(g))=1. This suggests that there are much more simple isogeny classes of abelian varieties over \mathbbFq of dimension g than non-simple ones for sufficiently large g, which can be understood as the opposite situation to a main result of Lipnowski and Tsimerman (Duke Math 167:3403-3453, 2018).