{
  "abstract": "In this article, we study the Mordell Weil ranks of elliptic curves in the family $E_D:y^2=x^3+Dx$, where $D\\in\\mathbb{Z}\\setminus\\{0\\}$. Under suitable congruence conditions on $D$ and its prime divisors, we use $2$-descent via a rational $2$-isogeny to study the rank of $E_D$. Under our congruence hypotheses, the ranks of these matrices give sufficient conditions for $E_D$ to have rank zero. We give explicit sufficient conditions for the curves $E_D$ to have Mordell Weil rank zero over $\\mathbb{Q}$ in the following cases: (i) $D$ is an odd negative integer, (ii) $D$ is an odd positive integer, and (iii) $D$ is a positive even integer. Also we give a quantitative result describing how many square free parameters satisfy the required matrix-rank condition. In particular, our results show that these rank zero criteria hold for a significant proportion of the admissible parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08029",
  "author_total": 3,
  "authors": [
    {
      "name": "Shamik Das"
    },
    {
      "name": "Debajyoti De"
    },
    {
      "name": "Arup Mondal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4exm8b8n9z8z1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08029",
  "fulltext_url": null,
  "id": "4exm8b8n9z8z1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08029",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4exm8b8n9z8z1.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Sufficient Conditions for Rank Zero in Families of Elliptic Curves",
  "title_text": "On Sufficient Conditions for Rank Zero in Families of Elliptic Curves",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/4exm8b8n9z8z1",
  "venue": "arXiv",
  "versions": []
}