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

$$\begin{aligned} \limsup _{g \rightarrow \infty }\frac{s_q(g)}{m_q(g)}=1. \end{aligned}$$

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).