{
  "abstract": "We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi--Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger--Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner--Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger--Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner--Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the $p$-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo $p^{2s}$ for every $s\\geq1$. Moreover, if the hypotheses of the Beukers--Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek--Verrill $A_4$ family at $t_*=-1/7$, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes $t_*$, the supercongruences hold, and the fiber with parameter $t_*\\bmod p^2$ is the Achinger--Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the $q$-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of $W(\\mathbb{F}_q)[1/p]$, where $q$ is the cardinality of the residue field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36045",
  "author_total": 1,
  "authors": [
    {
      "name": "Yu Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/34dd6szhx14xy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36045",
  "fulltext_url": null,
  "id": "34dd6szhx14xy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36045",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214965267",
  "pdf_url": "https://arxiv.org/pdf/2609.36045",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/34dd6szhx14xy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36045",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36045"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "p-adic Rigidity and Supercongruences at Rank Two Attractors",
  "title_text": "p-adic Rigidity and Supercongruences at Rank Two Attractors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/34dd6szhx14xy",
  "venue": "arXiv",
  "versions": []
}