{
  "abstract": "This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime $\\ell$, the finite unramified coverings $[\\ell^n]:E\\to E$ form a compatible tower whose deck groups are $E[\\ell^n]$; their inverse limit is the Tate module $T_\\ell(E)$, and over $\\Q$ the compatible Galois action gives the usual $\\ell$-adic representation. Cyclic isogenies, by contrast, give the index-$q$ lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve $E/\\Q$ of prime conductor $p$, the missing reciprocity is the construction, without using modularity, of a nonzero class $c_E$ in the degree-zero supersingular Brandt module such that $B_qc_E=a_q(E)c_E$ at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants $D_q=a_q(E)^2-4q$. New elementary results show that $D_q$ is a nonsquare modulo $p$ exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which $p$ is inert in the quadratic field determined by $D_q$, and that the corresponding discriminants are necessarily unbounded.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26196",
  "author_total": 1,
  "authors": [
    {
      "name": "Alberto Verjovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08h6mtytq3m27.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26196",
  "fulltext_url": null,
  "id": "08h6mtytq3m27",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26196",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214085102",
  "pdf_url": "https://arxiv.org/pdf/2609.26196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08h6mtytq3m27.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Peels of Equilateral Triangulated Surfaces",
  "title_text": "Peels of Equilateral Triangulated Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/08h6mtytq3m27",
  "venue": "arXiv",
  "versions": []
}