{
  "abstract": "Let $\\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\\mathop{\\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $RΓ(X_C,\\mathbb{Q}_{\\ell})$ is independent of $\\ell$ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i(X_C,\\mathbb{Q}_{\\ell})$, proving a conjecture of Lu--Zheng. In the general case, we give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i_c(X_C,\\mathbb{Q}_{\\ell})$, and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu--Zheng and Mieda to the positive characteristic case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17852",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiangdong Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bvfce6v879gx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17852",
  "fulltext_url": null,
  "id": "0bvfce6v879gx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17852",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221747599",
  "pdf_url": "https://arxiv.org/pdf/2610.17852",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bvfce6v879gx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.17852",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17852"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$\\ell$-Independence, slopes, and weights of $\\ell$-adic cohomology groups of rigid spaces",
  "title_text": "ℓ-Independence, slopes, and weights of ℓ-adic cohomology groups of rigid spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/0bvfce6v879gx",
  "venue": "arXiv",
  "versions": []
}