{
  "abstract": "We give explicit annular normal forms and classify Frobenius structures for $p$-adic Schwarzian equations defined by developing differentials. For a unit of the Robba ring, reduction of order expresses the associated module as a quadratic rank-one twist of a unipotent extension determined by the residue. Frobenius existence is characterized by a square-class condition. At odd primes, there are exactly four differential-module types, and a uniform dominant-monomial hypothesis gives a Frobenius formula in analytic families across the zero-residue locus. For rational nonintegral annular powers, we obtain a diagonal normal form and a congruence criterion at odd primes; both results hold at every prime when the unit factor is an explicit square. Applied to the reducible modular equations $θ^2y-\\ell^2E_4y/144=0$, with $\\gcd(\\ell,6)=1$, this gives exactly two annular types at every prime and determines the least $s\\geq1$ for which Frobenius under $q\\mapsto q^{p^s}$ exists. The subfamily $\\ell=12n+1$ has a single annular type. Heine--Stieltjes residue cancellation and nondegenerate Jacobi configurations provide explicit examples. We also prove an obstruction to constant projective Frobenius equivariance of Robba-ring developing maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21213",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohammadreza Mohajer"
    },
    {
      "name": "Abdellah Sebbar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjkqymb8ystj8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21213",
  "fulltext_url": null,
  "id": "bjkqymb8ystj8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21213",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213869741",
  "pdf_url": "https://arxiv.org/pdf/2609.21213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjkqymb8ystj8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations",
  "title_text": "Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/bjkqymb8ystj8",
  "venue": "arXiv",
  "versions": []
}