{
  "abstract": "Let $p\\geq 5$ be a prime, and let $f$ be a $p$-ordinary cuspidal newform of even weight $k = 2r \\geq 2$ and level $N$, with $p \\nmid N$ and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for $f$, formulated in terms of the $p$-adic $L$-function of $f$, under the sole assumptions that the associated Galois representation of $f$ satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale--Castella--Skinner, which treats elliptic curves over $\\mathbb{Q}$ with good ordinary reduction at $p$. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which $p$ splits, of a zeta element attached to $f$, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale--Skinner--Tian--Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale--Castella--Skinner and allow their method to be carried out for higher-weight modular forms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38558",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruichen Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1y9zktpa93014.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38558",
  "fulltext_url": null,
  "id": "1y9zktpa93014",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38558",
  "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": "W7215065954",
  "pdf_url": "https://arxiv.org/pdf/2609.38558",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1y9zktpa93014.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38558",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38558"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zeta elements for $p$-ordinary modular forms over imaginary quadratic fields and applications",
  "title_text": "Zeta elements for p-ordinary modular forms over imaginary quadratic fields and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1y9zktpa93014",
  "venue": "arXiv",
  "versions": []
}