On a number of isogeny classes of simple abelian varieties over finite fields

摘要

In this paper, we investigate the asymptotic behavior of the number sq(g). 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 Fq. We also prove that lim supg ->infinity sq(g)mq(g)=1. This suggests that there are much more simple isogeny classes of abelian varieties over Fq 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).