{
  "abstract": "We establish lower bounds on the orbit cardinalities of $\\text{SL}(2,\\mathbb{Z})$ on the $\\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = τ+2$. We show that outside of the reductions of a certain exceptional finite set of $\\overline{\\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\\text{SL}(2,\\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $τ= -2$. Our method studies certain noncongruence modular curves attached to a $\\text{SL}(2,\\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\\text{PSL}(2,\\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\\text{PSL}(2,\\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16407",
  "author_total": 1,
  "authors": [
    {
      "name": "William Y. Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/adgy1rkh2g29a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16407",
  "fulltext_url": null,
  "id": "adgy1rkh2g29a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16407",
  "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.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221804322",
  "pdf_url": "https://arxiv.org/pdf/2610.16407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/adgy1rkh2g29a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Stable reduction and large Nielsen orbits on generalized Markoff surfaces",
  "title_text": "Stable reduction and large Nielsen orbits on generalized Markoff surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/adgy1rkh2g29a",
  "venue": "arXiv",
  "versions": []
}