{
  "abstract": "Let $B$ be a quaternion division algebra over a number field $k$ that gives rise to arithmetic Fuchsian or Kleinian groups, and let $\\mathcal{O}$ be a maximal order of $B$. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface $X_\\mathcal{O}(\\mathfrak{a})$ is isometric to $X_{\\mathcal{O}'}(\\mathfrak{a})$ for some maximal order $\\mathcal{O}'$ of $B$, provided that no prime factor of $\\mathfrak{a}$ ramifies in $B$ and that $\\mathfrak{a}$ satisfies a mild condition at the primes over $2$. This is a converse to the construction of Vignéras, which produces isospectral manifolds from maximal orders of different types. In particular the spectral genus of $X_\\mathcal{O}(\\mathfrak{a})$ has at most $t(B)$ elements, and $X_\\mathcal{O}(\\mathfrak{a})$ is spectrally rigid when the type number $t(B)$ is one. In dimension three we prove the same statements for manifolds that, in addition, have the same complex lengths of closed geodesics. The proof rests on a local result that characterizes, by traces alone, the subgroups of $\\mathrm{SL}(2)$ over a $p$-adic ring that are conjugate into a principal congruence subgroup by traces. As applications, every principal congruence Hurwitz surface is spectrally rigid, among them Klein's quartic, the Fricke-Macbeath surface and the surfaces of the first Hurwitz triplet, which partially answers a question of Reid.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39874",
  "author_total": 2,
  "authors": [
    {
      "name": "Justin Katz"
    },
    {
      "name": "D. B. McReynolds"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5adyvp0pzxt4z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39874",
  "fulltext_url": null,
  "id": "5adyvp0pzxt4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39874",
  "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.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39874",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5adyvp0pzxt4z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39874",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39874"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras",
  "title_text": "Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5adyvp0pzxt4z",
  "venue": "arXiv",
  "versions": []
}