{
  "abstract": "Let \\(X\\subset \\mathbb P^n_{\\mathbf w}\\) be a weighted hypersurface over \\(\\mathbb F_q\\), with associated stack \\(\\mathfrak X\\). For \\(\\mathbf x\\in X(\\mathbb F_{q^r})\\), let \\(g_{\\mathbf x}(r)=\\gcd(k_{S(\\mathbf x)},q^r-1)\\), where \\(k_S=\\gcd(w_i:i\\in S)\\), and define \\(Θ_r(s)=\\sum_{\\mathbf x}g_{\\mathbf x}(r)^{1-s}\\) and \\(Z_H^{\\mathrm{can}}(\\mathfrak X,s;t)=\\exp(\\sum_{r\\ge1}Θ_r(s)t^r/r)\\). We prove the finite sector factorization \\[ Z_H^{\\mathrm{can}}(\\mathfrak X,s;t)=\\prod_{e\\in E_{\\mathbf w}} Z(X^{(e)}_{\\mathbb F_{q^{o_e}}},t^{o_e})^{J_{1-s}(e)/o_e}, \\] where \\(X^{(e)}\\) is the closed isotropy sector, \\(o_e=\\operatorname{ord}_e(q)\\) for \\(e>1\\) and \\(o_1=1\\), and \\(J_{1-s}\\) is the Jordan totient function. Hence the specializations \\(s=1,0,-1,-2,\\ldots\\) are rational in \\(t\\): \\(s=1\\) gives the Hasse--Weil zeta function of the coarse space, \\(s=0\\) the twist zeta function, and \\(s=1-k\\) the masses of the \\(k\\)-fold inertia stack. The factorization yields a functional equation when the nonempty sectors are self-dual and equidimensional; if they are also pure and geometrically irreducible, equidimensionality is necessary for every integral \\(s\\le0\\). For weighted diagonal hypersurfaces of degree \\(D\\) with \\(p\\nmid D\\), a finite Fermat cover gives purity and self-duality. We also prove a Lang--Weil asymptotic for isotropy entropy, with periodic leading term governed by isotropy periods and Frobenius orbits, and identify the function-field height with the stable stack height of Ellenberg--Satriano--Zureick-Brown, while the degree-based height remains a coordinate-complexity statistic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33162",
  "author_total": 2,
  "authors": [
    {
      "name": "S. Salami"
    },
    {
      "name": "T. Shaska"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2g76t9j73j96m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33162",
  "fulltext_url": null,
  "id": "2g76t9j73j96m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33162",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214794914",
  "pdf_url": "https://arxiv.org/pdf/2609.33162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2g76t9j73j96m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields",
  "title_text": "Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2g76t9j73j96m",
  "venue": "arXiv",
  "versions": []
}