Estimates for the number of rational points on simple abelian varieties over finite fields

Abstract

Let A be a simple Abelian variety of dimension g over the field $\mathbb {F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb {F}_q)$. For an arbitrary value of q we prove $(\lfloor (\sqrt{q}-1)^2 \rfloor + 1)^g \le \#A(\mathbb {F}_q) \le (\lceil (\sqrt{q}+1)^2 \rceil - 1)^{g}$ holds with finitely many exceptions. We compute improved bounds for various small values of q. For instance, the Weil bounds for $q=3,4$ give a trivial estimate $\#A(\mathbb {F}_q) \ge 1$; we prove $\# A(\mathbb {F}_3) \ge 1.359^g$ and $\# A(\mathbb {F}_4) \ge 2.275^g$ hold with finitely many exceptions. We use these results to give some estimates for the size of the rational 2-torsion subgroup $A(\mathbb {F}_q)[2]$ for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.