Vanishing of degree $3$ unramified cohomology over finite fields

Abstract

Let $k$ be a finite field of characteristic $p\ne 3$, let $E/k$ be the Fermat cubic curve, and let $\ell \ne p$ be a prime. If $p\equiv 1(\mathrm{mod}3)$, assume moreover that $\ell >3$. Then $H_{\mathrm{nr}}^3(k(E^3)/k,{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell }(2))=0$ and the cycle map \[CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,\mathbb{Z}_\ell(2))\] is surjective. In particular, $H_{\mathrm{nr}}^3(\overline{k}(E^3)/\overline{k},{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell }(2))=0$. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes $\ell \ne p$, 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.

arXiv comment54 pages