{
  "abstract": "Let $K$ be an imaginary quadratic field and let $p \\ge 5$ be a prime that is unramified in $K$. Let $\\mathcal{A}_f/\\mathbb{Q}$ be an abelian variety of $\\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In the case where $p$ is inert in $K$, we further assume that $\\mathcal{A}_f$ is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic $\\mathbb{Z}_p$-extensions that applies when the corresponding Selmer modules have arbitrary $Λ$-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale--Castella--Grossi--Skinner, Castella--Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13088",
  "author_total": 2,
  "authors": [
    {
      "name": "Antonio Lei"
    },
    {
      "name": "Luochen Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgz9nfbqy419r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13088",
  "fulltext_url": null,
  "id": "dgz9nfbqy419r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13088",
  "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/2609.13088",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgz9nfbqy419r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13088",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13088"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Kolyvagin's conjecture at non-ordinary primes",
  "title_text": "Kolyvagin's conjecture at non-ordinary primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/dgz9nfbqy419r",
  "venue": "arXiv",
  "versions": []
}