{
  "abstract": "We study the arithmetic of the elliptic curves \\[ A_m:y^2=x(x-m)(x+4m) \\] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\\equiv11\\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and show that the next Cassels--Tate pairing is governed by the normalized representation defect, equivalently by a factorial character, a Pell symbol, and a class number congruence. For composite parameters we compute the ordinary Cassels matrices of four twists and, in the two prime case, a degree $1024$ governing field for their joint distribution. The same higher descent extends to larger radicals: a second rational pushout determines the full $Λ'$ row of the next pairing. We also determine two explicit Selmer towers with four dimensional ordinary radical, and all finite $2$-power Selmer groups when the ordinary radical is one dimensional.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23419",
  "author_total": 2,
  "authors": [
    {
      "name": "Kaisheng Lei"
    },
    {
      "name": "Shisong Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e6292x8ram2cc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23419",
  "fulltext_url": null,
  "id": "e6292x8ram2cc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23419",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214008951",
  "pdf_url": "https://arxiv.org/pdf/2609.23419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e6292x8ram2cc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23419"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem",
  "title_text": "Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/e6292x8ram2cc",
  "venue": "arXiv",
  "versions": []
}