{
  "abstract": "Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33912",
  "author_total": 2,
  "authors": [
    {
      "name": "Ahmad Kazemifard"
    },
    {
      "name": "Saeed Tafazolian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nhqkywh1n11g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33912",
  "fulltext_url": null,
  "id": "3nhqkywh1n11g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33912",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214757875",
  "pdf_url": "https://arxiv.org/pdf/2609.33912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nhqkywh1n11g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the generalized Ree curve",
  "title_text": "On the generalized Ree curve",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3nhqkywh1n11g",
  "venue": "arXiv",
  "versions": []
}