{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36188",
  "author_total": 2,
  "authors": [
    {
      "name": "Federico Scavia"
    },
    {
      "name": "Fumiaki Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/feas75wzs8cb5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36188",
  "fulltext_url": null,
  "id": "feas75wzs8cb5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36188",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214932963",
  "pdf_url": "https://arxiv.org/pdf/2609.36188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/feas75wzs8cb5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Vanishing of degree $3$ unramified cohomology over finite fields",
  "title_text": "Vanishing of degree $3$ unramified cohomology over finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/feas75wzs8cb5",
  "venue": "arXiv",
  "versions": []
}