On the lower bound of the number of abelian varieties over ${\mathbb{F}}_p$

In this paper, we prove that the number $B(p,g)$ of isomorphism classes of abelian varieties over a prime field ${\mathbb{F}}_p$ of dimension $g$ has a lower bound $p^{\frac{1}{2}{g}^2(1+o(1))}$ as $g\to \mathrm{\infty }$. This is the first nontrivial result on the lower bound of $B(p,g)$. We also improve the upper bound $2^{34g^2}{p}^{\frac{69}{4}{g}^2(1+o(1))}$ of $B(p,g)$ given by Lipnowski and Tsimerman (Duke Math. J. 167:3403-3453, 2018) to $p^{\frac{45}{4} g^2(1+o(1))}$.