How Large is $A_g({\mathbb{F}}_q)$?

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p\mathrm{.}$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties over the finite field of size $p\mathrm{.}$ We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound $B(g,p)$ implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.