{
  "abstract": "For integers $t\\geq -1$, let $θ_t$ be the largest real root of the Shanks polynomial $g_t(X)=X^3-tX^2-(t+3)X-1$ and put $K_t=\\mathbb{Q}(θ_t)$. We classify the points on $E_n:y^2=x^3-n^2x$ over $K_t$ with abscissa $θ_t-1$ when $n$ is a positive integer and $E_n(\\mathbb{Q})$ has rank zero. We determine the four pairs $(d,t)$ with $d>0$ rational for which $(θ_t-1)/d^2$ is an abscissa on $E_3(K_t)$. We also exclude the abscissa $θ_t-1$ on $E_5(K_t)$ and $E_6(K_t)$ and prove that only finitely many parameters $t$ admit this abscissa for each fixed positive integer $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08398",
  "author_total": 1,
  "authors": [
    {
      "name": "Junyu Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q4a8es9a9j4v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08398",
  "fulltext_url": null,
  "id": "0q4a8es9a9j4v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08398",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212082820",
  "pdf_url": "https://arxiv.org/pdf/2609.08398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q4a8es9a9j4v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.08398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields",
  "title_text": "Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0q4a8es9a9j4v",
  "venue": "arXiv",
  "versions": []
}