{
  "abstract": "We show that the two modular curves of level $49$ and genus $9$ left open by the work of Rouse, Sutherland and Zureick-Brown, and of Furio and Lombardo, have no non-CM rational points. Together with the theorem of Furio and Lombardo on $X_{ns}^{+}(49)$, this completes the classification of the $7$-adic images of Galois of non-CM elliptic curves over $Q$. The proof is a Kummer descent on the superelliptic equations $F(x,y) = k\\,w^{7}$ of Furio and Lombardo, in the cases $7 \\mid k$ that they left open. The covering curves are twists of the Fermat septic, they map to twists of the Klein quartic, and a $7$-adic computation shows that the twist attached to a solution is trivial. The rational points of the Klein quartic were determined by Hurwitz, who reduced the question to Fermat's Last Theorem for exponent $7$, proved by Lamé. Thus, the last open case of the $7$-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent $7$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06521",
  "author_total": 2,
  "authors": [
    {
      "name": "Santiago Arango-Piñeros"
    },
    {
      "name": "David Zureick-Brown"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/73bsacpf8e2we.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06521",
  "fulltext_url": null,
  "id": "73bsacpf8e2we",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06521",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220670433",
  "pdf_url": "https://arxiv.org/pdf/2610.06521",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/73bsacpf8e2we.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06521",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06521"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent",
  "title_text": "$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/73bsacpf8e2we",
  "venue": "arXiv",
  "versions": []
}