  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.
