{
  "abstract": "For a fixed prime $p$, let $μ_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $μ_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin--Lehner subspaces $S_k^σ(N) \\subseteq S_k(N)$ and $S_k^{\\operatorname{new}, σ}(N) \\subseteq S_k^{\\operatorname{new}}(N)$. We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin--Serre conjecture to the Atkin--Lehner setting. For the second application, we obtain explicit bounds on the number of newforms $f \\in S_k^{\\operatorname{new}, σ}(N)$ for which $p$ is extremal over $S_k^{\\operatorname{new}, σ}(N)$. For the third application, we prove explicit asymptotics for the number of $\\mathbb{F}_{p^r}$-points on the modular Jacobian $J_0(N),$ as well as on its factors $J_0^{\\operatorname{new}}(N),$ $J_0^σ(N),$ and $J_0^{\\operatorname{new}, σ}(N)$. We also make explicit an asymptotic result of Serre concerning point counts of the modular curves $X_0(N)$. For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large $\\mathbb{Q}$-simple factors of $J_0(N)$ to analogous bounds for $J_0^σ(N)$. Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17806",
  "author_total": 4,
  "authors": [
    {
      "name": "Aarya J. Kumar"
    },
    {
      "name": "Sargam Mondal"
    },
    {
      "name": "Erick Ross"
    },
    {
      "name": "Hui Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2v7k9fs0mcbqc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17806",
  "fulltext_url": null,
  "id": "2v7k9fs0mcbqc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17806",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7213498627",
  "pdf_url": "https://arxiv.org/pdf/2609.17806",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2v7k9fs0mcbqc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17806",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17806"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces",
  "title_text": "Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/2v7k9fs0mcbqc",
  "venue": "arXiv",
  "versions": []
}