Non-algebraic geometrically trivial cohomology classes over finite fields

Abstract

We give the first examples of smooth projective varieties X over a finite field $F$ admitting a non-algebraic torsion ℓ-adic cohomology class of degree 4 which vanishes over $\bar{F}$. We use them to show that two versions of the integral Tate conjecture over $F$ are not equivalent to one another and that a fundamental exact sequence of Colliot-Thélène and Kahn does not necessarily split. Some of our examples have dimension 4, and are the first known examples of fourfolds with non-vanishing $H_{\mathrm{nr}}^3(X,Q_2/Z_2(2\left)\right)$.