{
  "abstract": "In this second chapter of the $\\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\\to\\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\\mathbb{P}^1\\to\\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29836",
  "author_total": 1,
  "authors": [
    {
      "name": "Francesco Naccarato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66v40r93kwrpp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29836",
  "fulltext_url": null,
  "id": "66v40r93kwrpp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29836",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204864435",
  "pdf_url": "https://arxiv.org/pdf/2608.29836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66v40r93kwrpp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2608.29836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The arithmetic of critical values II: critical elliptic curves",
  "title_text": "The arithmetic of critical values II: critical elliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/66v40r93kwrpp",
  "venue": "arXiv",
  "versions": []
}