{
  "abstract": "For every prime power $q$ and integer $n\\ge1$, we give a single Kummer equation defining an $\\mathbb{F}_{q^{2n}}$-maximal curve $\\mathcal X_{q,n}$. The family recovers the Hermitian curves for $n=1,2$, the Giulietti--Korchmáros curve for $n=3$, and the Beelen--Montanucci curves for odd $n\\ge5$. For $q>2$ and even $n\\ge4$, it gives new maximal curves of genus $(q^2-1)q^n/2$ that are not covered by the Hermitian curve over $\\mathbb{F}_{q^{2n}}$. We compute the genus for both parities of $n$ and give a uniform proof of maximality by counting rational places. For $q>2$ and odd $n\\ge5$, we remove the Galois hypothesis from the known non-covering results for the Beelen--Montanucci and Garcia--Güneri--Stichtenoth curves. For even $n\\ge4$, we determine the full automorphism group,of order $(q^n+1)q(q^2-1)$, and give explicit equations and genera for all Galois quotients, with closed-form equations for several classes of subgroups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19546",
  "author_total": 2,
  "authors": [
    {
      "name": "Liming Ma"
    },
    {
      "name": "Yipeng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9k3xjq7f9b7a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19546",
  "fulltext_url": null,
  "id": "d9k3xjq7f9b7a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19546",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213598202",
  "pdf_url": "https://arxiv.org/pdf/2609.19546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9k3xjq7f9b7a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A new family of maximal curves not covered by the Hermitian curve",
  "title_text": "A new family of maximal curves not covered by the Hermitian curve",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/d9k3xjq7f9b7a",
  "venue": "arXiv",
  "versions": []
}