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

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.

38 pages

References in corpus (2)

• A heuristic for the distribution of point counts for random curves over a finite field
• Orbital integrals and Dedekind zeta functions

Cited by in corpus (1)

• On a number of isogeny classes of simple abelian varieties over finite fields