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.