{
  "abstract": "Using the theory of von Neumann algebras, Vaughan Jones showed abstractly that there is a bridge between holomorphic cusp forms and random matrices. This bridge is supplied by a distinguished function he called a {\\it tracelike vector}. He posed the following problem: construct a tracelike vector to make this bridge explicit. We carry out Jones' vision for the full modular group. We construct holomorphic tracelike vectors for every finite-covolume Fuchsian group throughout the permissible Bergman range. For the modular group at the critical parameter $s=13$, we use the resulting unit tracelike vector to assign explicit group coefficients to cusp form multiplication operators, and form positive finite random matrices from a shared Haar-distributed unitary matrix. Almost surely, their normalised traces recover all joint operator moments, first as the matrix size tends to infinity, and then as the group-word cutoff tends to infinity. We also identify the limiting mixed second moment with a series of Petersson norms that involve Maass-Shimura operators of cusp forms. The exact nature of our use of AI is detailed in the introduction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39904",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikolaos Diamantis"
    },
    {
      "name": "Larry Rolen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2g28c9j2t2v74.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39904",
  "fulltext_url": null,
  "id": "2g28c9j2t2v74",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39904",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7215123310",
  "pdf_url": "https://arxiv.org/pdf/2609.39904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2g28c9j2t2v74.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2609.39904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras",
  "title_text": "On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/2g28c9j2t2v74",
  "venue": "arXiv",
  "versions": []
}