{
  "abstract": "Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)=\\sum_{i=0}^{h}a_ix^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\\Cart$ on $H^0(C_R,\\Omg)$. For $1\\le r\\le h-1$ the image of $\\Cart^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\\rank(\\Cart^r)=p^{h-r-1}\\binom{p+r}{r+2}$. The last iterate detects the linear coefficient: $\\rank(\\Cart^h)=0$ if $a_0=0$ and $\\rank(\\Cart^h)=\\binom{(p-1)/2+h}{h+1}$ if $a_0\\neq0$, so the nilpotency index of $\\Cart$ is $h$ or $h+1$. Hence, for $h\\ge2$, the loci $a_0=0$ and $a_0\\ne0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over $\\F_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be $\\F_{p^{2n}}$-maximal or $\\F_{p^{2n}}$-minimal only if $n\\ge h$, and only if $n\\ge h+1$ when $a_0\\neq0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17476",
  "author_total": 3,
  "authors": [
    {
      "name": "Gilberto B. Almeida Filho"
    },
    {
      "name": "Saeed Tafazolian"
    },
    {
      "name": "Stéfani C. Vieira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ff0x6372epbs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17476",
  "fulltext_url": null,
  "id": "8ff0x6372epbs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17476",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221817948",
  "pdf_url": "https://arxiv.org/pdf/2610.17476",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ff0x6372epbs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17476",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17476"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$a$-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves",
  "title_text": "a-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/8ff0x6372epbs",
  "venue": "arXiv",
  "versions": []
}