{
  "abstract": "Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $π$ be an irreducible smooth depth-zero representation of $G(E)$ over $\\overline{\\mathbb{F}}_{\\ell}$ where $\\ell$ is a prime number which does not divide $p|π_0Z(G)|$. Let $φ_π^{\\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $π$. We describe the restriction of $φ_π^{\\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues--Scholze $L$-parameters agree with DeBacker--Reeder $L$-parameters over $\\overline{\\mathbb{Q}}_{\\ell}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16445",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenji Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8v5zeqpx3bd4f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16445",
  "fulltext_url": null,
  "id": "8v5zeqpx3bd4f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16445",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221771856",
  "pdf_url": "https://arxiv.org/pdf/2610.16445",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8v5zeqpx3bd4f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16445",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16445"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Depth-zero local Langlands via shtukas in mixed characteristic",
  "title_text": "Depth-zero local Langlands via shtukas in mixed characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8v5zeqpx3bd4f",
  "venue": "arXiv",
  "versions": []
}