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

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 $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g \leqslant A(\mathbb{F}_q) \leqslant (\lceil(\sqrt{q}+1)^2 \rceil - 1)^{g}$ 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 $A(\mathbb{F}_q) \geqslant 1$; we prove $A(\mathbb{F}_3) \geqslant 1.359^g$ and $A(\mathbb{F}_4) \geqslant 2.275^g$ hold with finitely many exceptions. We use these results to describe all abelian varieties over finite fields that have no new points in some finite field extension.