{
  "abstract": "Let $K/\\mathbb Q$ be an imaginary quadratic extension and $p$ an odd prime. Write $ρ=\\operatorname{Ind}_{G_K}^{G_{\\mathbb Q}}χ$, where $E/\\mathbb Q_p$ is a finite extension and $χ:G_K\\to\\mathcal O_E^\\times$ is a continuous character. For a fixed $r\\in\\mathbb Z\\setminus\\{0\\}$, let $π_{ρ,r}(X)$ denote the number of rational primes $\\ell\\le X$ such that $ρ$ is unramified at $\\ell$ and $\\operatorname{tr}ρ(\\operatorname{Frob}_\\ell)=r$. Let $a,b$ be the two weights of $χ$ at $p$. We prove that if $(a,b)\\notin\\mathbb Q^2$, then $π_{ρ,r}(X)\\ll_{ρ,r,\\varepsilon}X^\\varepsilon$ for every $\\varepsilon>0$, while if $(a,b)\\in\\mathbb Q^2\\setminus\\mathbb Z^2$, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang--Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the \\(p\\)-adic analytic trace locus, combined with rigid-analytic Pila--Wilkie counting; the rational non-integral case is treated by a local ramification argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08201",
  "author_total": 1,
  "authors": [
    {
      "name": "Haoyang Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csa4jegskt6tv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08201",
  "fulltext_url": null,
  "id": "csa4jegskt6tv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08201",
  "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": "W7212062757",
  "pdf_url": "https://arxiv.org/pdf/2609.08201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csa4jegskt6tv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08201"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Lang-Trotter Problem for Non-Geometric Quadratic Inductions",
  "title_text": "A Lang-Trotter Problem for Non-Geometric Quadratic Inductions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/csa4jegskt6tv",
  "venue": "arXiv",
  "versions": []
}