{
  "abstract": "For an odd prime p, we study the change in matrix order under lifting from Z/p^k Z to Z/p^(k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl_2(F_p). We classify the full-order transfer for every q in GL_2(F_p): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U^2 = 0 and U != 0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX) != 0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruction criterion for order preservation among unrestricted Jordan lifts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18900",
  "author_total": 1,
  "authors": [
    {
      "name": "Ivo Vajc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9k0e79wdw3ekn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18900",
  "fulltext_url": null,
  "id": "9k0e79wdw3ekn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18900",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7221950544",
  "pdf_url": "https://arxiv.org/pdf/2610.18900",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9k0e79wdw3ekn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18900",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18900"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Rings, Modules, and Algebras",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings",
  "title_text": "Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/9k0e79wdw3ekn",
  "venue": "arXiv",
  "versions": []
}