See survey in the Motives volumes.
http://mathoverflow.net/questions/73270/tate-conjecture-for-elliptic-curves-local-fields
An interesting article by Kahn
arXiv:1101.1730 On the generalised Tate conjecture for products of elliptic curves over finite fields from arXiv Front: math.AG by Bruno Kahn We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.
Andre: Cycles de Tate… proves something weaker (motivated cycles) for abelian varieties
arXiv:1206.4002 The Tate conjecture for K3 surfaces over finite fields from arXiv Front: math.AG by François Charles Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p>3. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p>3. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p>3.
nLab page on Tate conjecture