{
  "abstract": "Let $E/\\mathbf{Q}$ be an elliptic curve, let $K=\\mathbf{Q}(\\sqrt{D})$ satisfy the Heegner hypothesis with $D<-4$, and fix a modular parametrization $φ$ with $φ(\\infty)=O$. Under surjectivity modulo $2^L$, with $L\\ge 2$, and a cohomological exclusion condition at level $4$, we construct a finite set of primes $T$ such that, writing $d_T=\\prod_{q\\in T}(-q)$ and $n_T=\\prod_{q\\in T}q$, the $2$-Selmer dimensions of $E^{d_T}$ and $E^{d_TD}$ are $0$ and $1$. The support size is bounded in terms of the initial Selmer dimensions, independently of $L$. Inspired by the work of Tian and Tian--Yuan--Zhang on congruent numbers, we construct a point $Q_T$ on the terminal twist and identify its Kummer class with the standard mod-$2$ Kolyvagin class. Using the low-corank $2$-converse and the $2$-primary Birch--Swinnerton-Dyer formula, we prove that its index $m_T$ in the full Mordell--Weil lattice satisfies $v_2(m_T)=v_2(c_φ)+\\mathfrak{i}(E,K)$. The optimal Manin constant theorem supplies a $2$-indivisible parametrization $φ_0$, for which $κ^{\\mathrm{std}}_{n_T,1}(φ_0)\\ne 0$ if and only if $\\mathfrak{i}(E,K)=0$. When the intrinsic index vanishes, the minimum terminal support size is $R=\\min\\{d_2(E),d_2(E^D)\\}$, and the normalized standard classes have order $2^M$ for $1\\le M\\le L$. Full $2$-adic Galois image then gives strong nonvanishing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31631",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhiqi Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/50hzg3kn75nwd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31631",
  "fulltext_url": null,
  "id": "50hzg3kn75nwd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31631",
  "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": "W7223299755",
  "pdf_url": "https://arxiv.org/pdf/2610.31631",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/50hzg3kn75nwd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31631",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31631"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists",
  "title_text": "Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/50hzg3kn75nwd",
  "venue": "arXiv",
  "versions": []
}