发表机构
CNRS Institut Galilée Université Sorbonne Paris Nord; BICMR Peking University（法国国家科学研究中心伽利莱研究所索邦巴黎北大学; 北京大学北京数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限域上费马三次曲线三次非分岐上同调消没及循环映射满射，并推广至满足 Tate 猜想的簇，应用于零循环局部-全局原理。
AI 中文摘要
设 $k$ 为特征 $p\neq 3$ 的有限域，$E/k$ 为费马三次曲线，$\ell\neq p$ 为素数。若 $p\equiv1\pmod3$，进一步假设 $\ell>3$。则 $H^3_{\mathrm{nr}}(k(E^3)/k,\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$ 且循环映射 $CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,\mathbb{Z}_\ell(2))$ 是满射。特别地，$H^3_{\mathrm{nr}}(\overline{k}(E^3)/\overline{k},\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$。假设有限域上曲面的 Tate 猜想成立，我们证明了一个类似的满射性结果，对于所有但有限多个素数 $\ell\neq p$，针对特征不同于 $2$ 的有限域上每个光滑射影 $d$ 维簇上的 $1$-循环的积分循环映射，该簇承认到 Witt 向量环的光滑射影提升。我们将我们的结果应用于 Colliot-Thélène 关于全局函数域上零循环的局部-全局原理的猜想。为进一步说明这些结果，我们展示了例子，表明基域代数闭包上三次非分岐上同调的消没并不蕴含任何有限子扩张上的消没，甚至对 Fano 簇也不成立。
英文摘要
Let $k$ be a finite field of characteristic $p\neq 3$, let $E/k$ be the Fermat cubic curve, and let $\ell\neq p$ be a prime. If $p\equiv1\pmod3$, assume moreover that $\ell>3$. Then $H^3_{\mathrm{nr}}(k(E^3)/k,\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$ and the cycle map \[CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,\mathbb{Z}_\ell(2))\] is surjective. In particular, $H^3_{\mathrm{nr}}(\overline{k}(E^3)/\overline{k},\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes $\ell\neq p$, 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.
Comments54 pages
