  This is loosely a continuation of the author's previous paper
arXiv:1802.09496. In the first part, given a fibered variety, we pull back the
Leray filtration to the Chow group, and use this to give some criteria for the
Hodge and Tate conjectures to hold for such varieties. In the second part, we
show that the Hodge conjecture holds for a good desingularization of a self
fibre product of a non-isotrivial elliptic surface. We also show that the Hodge
and Tate conjectures hold for natural families of abelian varieties
parameterized by certain Shimura curves. This uses Zucker's description of the
mixed Hodge structure on the cohomology of a variation of Hodge structures on a
curve, along with appropriate "vanishing" theorems.
