  Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional
abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the
number of isomorphism classes of principally polarized $g$ dimensional abelian
varieties over the finite field of size $p.$ 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.
