{
  "abstract": "The Richelot $(2,2)$-isogeny graph $G_p$ of superspecial abelian surfaces in characteristic $p$ is the genus-2 analogue of the supersingular isogeny graph: its weighted adjacency matrix is a Brandt matrix for a compact inner form of $\\mathrm{GSp}_4$, and it commutes with an Atkin-Lehner involution $R(π)$. Refining Ibukiyama's conjectures, we conjecture the automorphic origin and the $R(π)$-sign of every eigenvalue: Eisenstein, Saito-Kurokawa, Yoshida and paramodular eigenvalues, and pairs of type-Va eigensystems that only $R(π)$ separates. We compute $G_p$ at 33 primes, with up to 2176 vertices, and certify the predicted eigenvalues and signs in exact arithmetic; at $p=167$ and $173$, where the sign prediction first becomes non-trivial, the graph carries the predicted eigenvalue $-8$ with sign $-1$. For every prime we prove that each type-Va parameter contributes exactly one eigenline of each $R(π)$-sign, a closed formula for the trace of $R(π)$, and a Fricke sign bias. If no further CAP representations occur, the non-trivial spectrum lies in $[-8\\sqrt{2}, 6+4\\sqrt{2}]$, both ends being sharp under the conjecture, and only Saito-Kurokawa eigenvalues exceed $8\\sqrt{2}$; at all 104 primes $p\\le 601$ the extreme eigenvalue computed by Florit and Smith agrees with the largest Saito-Kurokawa one. Finally, the irrational roots of the residual square factor form one Galois orbit with full symmetric Galois group at each certified prime $p\\ge 29$, and the spectrum appears to follow a Plancherel law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14145",
  "author_total": 1,
  "authors": [
    {
      "name": "Hung T. Dang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jgyxa2ctmba0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14145",
  "fulltext_url": null,
  "id": "8jgyxa2ctmba0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14145",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7203615426",
  "pdf_url": "https://arxiv.org/pdf/2608.14145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jgyxa2ctmba0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.14145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra",
  "title_text": "The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/8jgyxa2ctmba0",
  "venue": "arXiv",
  "versions": []
}