{
  "abstract": "We study the $θ$-congruent number problem for $\\cosθ=\\pm3/5$ and $\\pm4/5$ using the generalized theta series construction of Sirolli--Tornaría. We describe its specialization to newforms of weight $2$ over $\\mathbb Q$ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every $θ$-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight $3/2$ whose Fourier coefficients determine the central $L$-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-$θ$-congruence unconditionally, and the converse holds assuming the Birch--Swinnerton-Dyer conjecture. We also prove unconditional non-$θ$-congruence for primes in explicit arithmetic progressions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19085",
  "author_total": 2,
  "authors": [
    {
      "name": "Bo-Hae Im"
    },
    {
      "name": "Minseo Shin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftymnyhebew71.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19085",
  "fulltext_url": null,
  "id": "ftymnyhebew71",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19085",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213517585",
  "pdf_url": "https://arxiv.org/pdf/2609.19085",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftymnyhebew71.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.19085",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19085"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Tunnell-type criteria for variants of the congruent number problem",
  "title_text": "Tunnell-type criteria for variants of the congruent number problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/ftymnyhebew71",
  "venue": "arXiv",
  "versions": []
}