How large is Ag(Fq)?

Abstract

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 for $B(g,p)$ implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.