{
  "abstract": "We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \\ge 1$, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations $ρ: G_K \\to \\mathrm{GL}_n(\\overline{\\mathbf{F}}_p)$ that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields $K$ of characteristic $p$ admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation $ρ_r: G_K \\twoheadrightarrow \\mathrm{SL}_2(\\mathbf{F}_{p^r})$ for each integer $r \\ge 4$. As a corollary, we construct an everywhere unramified Galois extension \\(L/K\\), regular over \\(\\mathbf F_p\\), with Galois group \\[ \\mathrm{Gal}(L/K)\\cong \\prod_{r\\geq4}\\mathrm{PSL}_2(\\mathbf F_{p^r}). \\] Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic \\(p\\), whereas they cannot exist over global function fields of characteristic different from \\(p\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11456",
  "author_total": 2,
  "authors": [
    {
      "name": "Yufan Luo"
    },
    {
      "name": "Yiqi Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8evrxd1zj2tn6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11456",
  "fulltext_url": null,
  "id": "8evrxd1zj2tn6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11456",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7212294192",
  "pdf_url": "https://arxiv.org/pdf/2609.11456",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8evrxd1zj2tn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11456",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11456"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the bounded-conductor finiteness conjecture in equal characteristic",
  "title_text": "On the bounded-conductor finiteness conjecture in equal characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/8evrxd1zj2tn6",
  "venue": "arXiv",
  "versions": []
}