{
  "abstract": "We establish an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\\mathrm{GL}_{2}$ $L$-functions over an arbitrary number field. A Rankin--Selberg kernel gives a new analytic regularization of the formal Petersson identity for products of Eisenstein series, based on the rank decomposition $4=2+2$. The formula preserves the nonnegativity of the amplified fourth-moment spectral side at general levels. We further identify a new \\emph{square-level concentration phenomenon}: after amplification, the dominant off-diagonal contribution comes from representations whose conductor is the square of a product of two distinct primes. Using square-level local test functions in the relative trace formula, we prove new hybrid subconvexity bounds for twisted $L$-functions. Combining these bounds with symmetric spectral reciprocity and amplification, we obtain the uniform estimate \\begin{align*} L(1/2,π)\\ll_{F,\\varepsilon} C(π)^{\\frac{1}{4}-\\frac{1}{120}+\\varepsilon} \\end{align*} for every unitary cuspidal automorphic representation $π$ of $\\mathrm{GL}_{2}/F$ with arbitrary central character. We also obtain stronger bounds for fixed central characters, a prime-level fifth-moment estimate, and applications to Rankin--Selberg and ring class $L$-functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04476",
  "author_total": 1,
  "authors": [
    {
      "name": "Liyang Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1f5h5ktpm2cpf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04476",
  "fulltext_url": null,
  "id": "1f5h5ktpm2cpf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04476",
  "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": "W7167619909",
  "pdf_url": "https://arxiv.org/pdf/2607.04476",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1f5h5ktpm2cpf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:09Z",
      "native_id": "2607.04476",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04476"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Symmetric Spectral Reciprocity for $\\mathrm{GL}(2)$ and Uniform Subconvexity",
  "title_text": "Symmetric Spectral Reciprocity for GL(2) and Uniform Subconvexity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/1f5h5ktpm2cpf",
  "venue": "arXiv",
  "versions": []
}