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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5285