{
  "abstract": "Let $p$ be an odd prime and let $\\mathbf{G}/\\mathbb Z$ be a split connected reductive group with $\\operatorname{dim} Z(\\mathbf{G})\\leq1$. Assume that a split maximal torus of $\\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\\mathrm{Lie}(\\mathbf{G}^{\\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $ρ:G_{\\{p\\}}\\longrightarrow\\mathbf{G}(\\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\\mathbf G^{\\mathrm{der}}(\\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\\mathbf{G}(\\mathbb{Z}_p)$. The construction continues the author's earlier work on $\\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\\mathbf{G}=\\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\\{p\\}}\\to\\mathrm{GL}_n(\\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\\geq7$ in that construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03954",
  "author_total": 1,
  "authors": [
    {
      "name": "Anwesh Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fasp9ngn4fe8x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03954",
  "fulltext_url": null,
  "id": "fasp9ngn4fe8x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03954",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7208777178",
  "pdf_url": "https://arxiv.org/pdf/2609.03954",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fasp9ngn4fe8x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03954",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03954"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois representations ramified at one prime via relative deformation theory",
  "title_text": "Galois representations ramified at one prime via relative deformation theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/fasp9ngn4fe8x",
  "venue": "arXiv",
  "versions": []
}