{
  "abstract": "We establish a Petersson-Rankin-Selberg identity for Hilbert Rankin-Cohen brackets over totally real fields of degree greater than one and arbitrary narrow class number. This extends results of Zhang-Zhang and Zhang-Zhou. For even $k,\\ell\\ge 2$, $r\\in\\mathbb{Z}_{\\ge 0}$, $K=k+\\ell+2r$, and normalized full-level cuspidal Hecke eigentuples $f\\in S_K(ω_f)$ and $h\\in S_\\ell(ω_h)$, we show $\\langle f,[E_k,h]_r\\rangle=C_{F,k,\\ell,r}L^S(k/2,Π(f)\\timesΠ(h)^\\vee)/L_F(k,ω_fω_h^{-1})\\ne 0$, where $C_{F,k,\\ell,r}>0$. Specializing to $k=2$, we remove the remaining GRH assumption in the product classification of Hao-Qin-Zhou. Hence, over real quadratic fields of narrow class number one, the only full-level product identities among Hecke eigenforms of even parallel weights at least two, up to interchanging the factors, are $E_4^{\\mathrm{eig}}=60(E_2^{\\mathrm{eig}})^2$ and $h_8=120E_2^{\\mathrm{eig}}h_6$ over $\\mathbb{Q}(\\sqrt{5})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28315",
  "author_total": 1,
  "authors": [
    {
      "name": "Jialin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bx93wmmss389.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28315",
  "fulltext_url": null,
  "id": "9bx93wmmss389",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28315",
  "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": "W7214152994",
  "pdf_url": "https://arxiv.org/pdf/2609.28315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bx93wmmss389.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Rankin-Cohen Pairings and Hilbert Hecke Eigenform Product Identities",
  "title_text": "Rankin-Cohen Pairings and Hilbert Hecke Eigenform Product Identities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/9bx93wmmss389",
  "venue": "arXiv",
  "versions": []
}