The integral Tate conjecture over finite fields and two coniveau filtrations

We construct a new type of counterexamples to the integral Tate conjecture over finite fields, where a geometric cycle map is surjective but an arithmetic cycle map is not. We also discuss the relation of this problem with two coniveau filtrations, and show some positive results toward a conjecture of Colliot-Thélène and Kahn. This is joint work with Federico Scavia.