The Tate conjecture is a long-standing problem in arithmetic geometry, describing algebraic cycles on an algebraic variety in terms of Galois representation on étale cohomology. In contrast, an integral analogue of the Tate conjecture is known to fail in general, and a natural question is whether the failure is always caused by geometry. Over a finite field, we construct the first counterexamples to this question: in codimension 2 on our examples, a geometric cycle map is surjective but an arithmetic cycle map is not. We also show positive results toward a conjecture of Colliot-Thélène and Kahn on the third unramified cohomology group for threefolds over a finite field. This is a joint work with Federico Scavia.

Non-algebraic geometrically trivial cohomology classes over finite fields Sponsored by the Simons Foundation