{
  "abstract": "Let $K/E$ be a finite unramified extension of $p$-adic local fields, and let $H$ be a one-dimensional formal $\\mathcal{O}_E$-module of finite height over $\\mathcal{O}_K$. We introduce the exponential period map of $H$ and use it to construct a complete regular local ring $R_{H,K}$ with imperfect residue field, an endomorphism $φ_E$, and a commuting action of $Γ=\\mathrm{Gal}(K(H[p^\\infty](\\overline{K}))/K)$. We prove an equivalence of categories between finitely generated $\\mathcal{O}_E$-modules with a continuous $\\mathrm{Gal}_K$-action and étale $(φ_E,Γ)$-modules over $R_{H,K}$. This recovers the classical cyclotomic and Lubin-Tate equivalences in the corresponding cases. In general, $R_{H,K}$ can have Krull dimension greater than one, and $φ_E$ need not lift the $q_E$-power Frobenius modulo $π_E$. The proof uses $F$-dynamical systems over $\\mathcal{O}_E$, which combine contraction modulo $π_E$ with Frobenius on the residue field. For every flat $F$-dynamical system, we establish an equivalence of categories between étale $φ$-modules and continuous representations of the absolute Galois group of the residue field on finitely generated $\\mathcal{O}_E$-modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31608",
  "author_total": 1,
  "authors": [
    {
      "name": "Daishi Kiyohara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9sdabbzym60gp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31608",
  "fulltext_url": null,
  "id": "9sdabbzym60gp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31608",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214657971",
  "pdf_url": "https://arxiv.org/pdf/2609.31608",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9sdabbzym60gp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31608",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31608"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Formal groups and $(φ,Γ)$-modules",
  "title_text": "Formal groups and (φ,Γ)-modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/9sdabbzym60gp",
  "venue": "arXiv",
  "versions": []
}