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

Abstract: In this paper, we investigate the asymptotic behavior of the number $s_q(g)$ of isogeny classes of simple abelian varieties of dimension $g$ over a finite field $\mathbb{F}q$. We prove that the logarithmic asymptotic of $s_q(g)$ is the same as the logarithmic asymptotic of the number $m_q(g)$ of isogeny classes of all abelian varieties of dimension $g$ over $\mathbb{F}_q$. We also prove that $$ \limsup{g \rightarrow \infty} \frac{s_q(g)}{m_q(g)}=1. $$ This suggests that there are much more simple isogeny classes of abelian varieties over $\mathbb{F}_q$ 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).