  In two earlier articles, we proved that, if the Hodge conjecture is true for
ALL CM abelian varieties over the complex numbers, then both the Tate
conjecture and the standard conjectures are true for abelian varieties over
finite fields. Here we rework the proofs so that they apply to a single abelian
variety. As a consequence, we prove (unconditionally) that the Tate and
standard conjectures are true for many abelian varieties over finite fields,
including abelian varieties for which the algebra of Tate classes is not
generated by divisor classes.
