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

TL;DR

This paper investigates the growth of isomorphism classes of abelian varieties over finite fields, providing bounds that reveal surprising statistical behaviors as the dimension increases.

Contribution

It derives new upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$, highlighting the significant gap between these bounds for large $g$.

Findings

Upper bounds for $B(g,p)$ established

Lower bounds for $A(g,p)$ established

Large gap implies counterintuitive statistical behavior

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