Vanishing of degree $3$ unramified cohomology over finite fields

Let $k$ be a finite field of characteristic $p\mathrm{\ne }3$, let $E\mathrm{/}k$ be the Fermat cubic curve, and let $\mathrm{\ell }\mathrm{\ne }p$ be a prime. If $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, assume moreover that $\mathrm{\ell }>3$. Then $H_{\mathrm{n}\mathrm{r}}^3(k(E^3)\mathrm{/}k,{\mathbb{Q}}_{\mathrm{\ell }}\mathrm{/}{\mathbb{Z}}_{\mathrm{\ell }}(2))=0$ and the cycle map $$C{H}^2(E^3{)}_{{\mathbb{Z}}_{\mathrm{\ell }}}⟶H^4(E^3,{\mathbb{Z}}_{\mathrm{\ell }}(2))$$ is surjective. In particular, $H_{\mathrm{n}\mathrm{r}}^3(\bar{k}(E^3)\mathrm{/}\bar{k},{\mathbb{Q}}_{\mathrm{\ell }}\mathrm{/}{\mathbb{Z}}_{\mathrm{\ell }}(2))=0$. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes $\mathrm{\ell }\mathrm{\ne }p$, for the integral cycle maps for $1$-cycles on every smooth projective variety of dimension $d$ over a finite field of characteristic different from $2$ which admits a smooth projective lift to the ring of Witt vectors. We apply our results to a conjecture of Colliot-Thélène on the local–global principle for zero-cycles over global function fields. To further illustrate these results, we exhibit examples showing that vanishing of degree-$3$ unramified cohomology over the algebraic closure of the ground field does not imply vanishing over any finite subextension, not even for Fano varieties.