{
  "abstract": "Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\\Q$ containing $\\infty$ but not $p$, any continuous odd representation $G_{\\Q,S} \\to \\GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\\Q,S}$ denotes the Galois group of the maximal extension of $\\Q$ unramified outside $S$. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine--Mazur conjecture over $\\Q$. Furthermore, we show that for an imaginary quadratic field $K$, any continuous conjugate self-dual two-dimensional $p$-adic representation of its pro-$p$ Galois group $G_{K,S}(p)$ has finite image, provided the primes in $S$ satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's $(3,3)$-conjecture on $p$-class field towers for $p>3$. Namely, we prove that if $K$ is an imaginary quadratic field with $p$-class rank two and the Galois group $G_{K,\\varnothing}(p)$ of the maximal unramified $p$-extension of $K$ has Zassenhaus type $(3,3)$, then the $p$-class field tower of $K$ is finite. For $p=3$, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37252",
  "author_total": 1,
  "authors": [
    {
      "name": "Yufan Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8dsq1ndjykkq0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37252",
  "fulltext_url": null,
  "id": "8dsq1ndjykkq0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37252",
  "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.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214926831",
  "pdf_url": "https://arxiv.org/pdf/2609.37252",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8dsq1ndjykkq0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37252",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37252"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Boston's Unramified Conjecture for $\\GL_2$ and McLeman's $(3,3)$-Conjecture",
  "title_text": "On Boston's Unramified Conjecture for GL_2 and McLeman's (3,3)-Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/8dsq1ndjykkq0",
  "venue": "arXiv",
  "versions": []
}