{
  "abstract": "Let $p$ and $\\ell$ be distinct odd primes. For a finite extension $F/\\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\\text{GL}_n(F)$ and $n$-dimensional, $Φ$-semisimple Weil--Deligne representations of the Weil group $W_F$. Given two $2$-dimensional, semisimple, continuous representations $ρ_1, ρ_2$ of $W_F$ with images in $\\text{GL}_2(\\mathcal{O}_K)$, where $K/\\mathbb{Q}_\\ell$ is a finite extension with maximal ideal $λ\\subset \\mathcal{O}_K$, a natural question to ask is when their mod $λ$ reductions $\\overlineρ_1$ and $\\overlineρ_2$ are isomorphic. In this paper, we give a complete characterization of when the reductions are isomorphic. For this, we first give a description of when the reductions of these representations are decomposable or irreducible, utilizing the correspondence between continuous $\\ell$-adic representations of $W_F$ and $\\ell$-adic Weil--Deligne representations. We conclude with some examples which arise in the modular method for solving generalized Fermat equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36467",
  "author_total": 2,
  "authors": [
    {
      "name": "Imin Chen"
    },
    {
      "name": "Deniz Suozer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x6nbf9k9d3qx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36467",
  "fulltext_url": null,
  "id": "1x6nbf9k9d3qx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36467",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214989803",
  "pdf_url": "https://arxiv.org/pdf/2609.36467",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x6nbf9k9d3qx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36467",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36467"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Reduction of Weil-Deligne Representations",
  "title_text": "Reduction of Weil-Deligne Representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/1x6nbf9k9d3qx",
  "venue": "arXiv",
  "versions": []
}