{
  "abstract": "For a prime $p\\geq 5$, let $\\mathscr S_g$ be the moduli space over $\\overline{\\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\\{\\pm1\\}$: (i) certain supersingular Ekedahl--Oort strata when $g$ is even, (ii) the loci in $\\mathscr S_g$ with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when $g\\geq 3$, and (iii) the locus in $\\mathscr S_g$ with $a$-number at least $2$ when $g\\geq 4$. Consequently, for $g\\geq 4$, the complement in $\\mathscr S_g$ of the open locus where the automorphism group is $\\{\\pm1\\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\\geq 5$. We reduce them to statements about affine Deligne--Lusztig varieties for $\\operatorname{GSp}_{2g}$ and prove analogues of (ii) for $\\operatorname{GL}_{2g}$ and $\\operatorname{GSO}_{4m}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03188",
  "author_total": 2,
  "authors": [
    {
      "name": "Ryosuke Shimada"
    },
    {
      "name": "Teppei Takamatsu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8kgq62vp5k8sj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03188",
  "fulltext_url": null,
  "id": "8kgq62vp5k8sj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03188",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7208788835",
  "pdf_url": "https://arxiv.org/pdf/2609.03188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8kgq62vp5k8sj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Some generalizations of Oort's conjecture",
  "title_text": "Some generalizations of Oort's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8kgq62vp5k8sj",
  "venue": "arXiv",
  "versions": []
}