  The characteristic polynomials of abelian varieties over the finite field
$\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric
information. They have been explicitly described for abelian varieties up to
dimension 4, but little is known in higher dimension. In this paper, among
other things, we obtain the following three results on the characteristic
polynomial of abelian varieties. First, we prove a relation between $n$ and
$e$, where $e$ is a certain multiplicity associated with a simple abelian
variety of arbitrary dimension over $\mathbb{F}_q$. Second, we explicitly
describe the characteristic polynomials of simple abelian varieties of
arbitrary dimension $g$, when $e=g$. Finally, we explicitly describe the
coefficients of characteristic polynomials of abelian varieties of dimension 5
over $\mathbb{F}_q$.
