{
  "abstract": "For a prime $p$, a pair $(s,m)\\in\\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $χ:C_m\\rightarrow\\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\\rtimes_χ C_m$, the $p$-Sylow subgroup $H\\cong C_{p^s}$ of $G$, and a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $χ$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\\widetilde{V}$ and (ii) when we require $\\widetilde{V}^H=\\{0\\}$. The criteria correct several results in the literature and we use them to prove that, if $χ$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\\rightarrow X/G$ is a Harbater--Katz--Gabber cover, the $k[G]$-module $H^0(X,Ω_X)$ has a lift $\\widetilde{V}$ to $R$ with $\\widetilde{V}^H=\\{0\\}$. Also, with $B:=R[\\frac{1}{p}]$, we prove the following obstruction when $p$ is odd, $m$ is even, $χ$ is injective, and $X/C_{p^s}\\cong\\mathbb{P}^1_k$: if no such lift $\\widetilde{V}$ exists with the $B[H]$-module $\\widetilde{V}\\otimes_R B$ defined over $\\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03191",
  "author_total": 2,
  "authors": [
    {
      "name": "Huy Dang"
    },
    {
      "name": "Adrian Vasiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3dxm448exafgj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03191",
  "fulltext_url": null,
  "id": "3dxm448exafgj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03191",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7208750357",
  "pdf_url": "https://arxiv.org/pdf/2609.03191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3dxm448exafgj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.03191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On lifting representations and actions on curves of the metacyclic groups $C_{p^s}\\rtimes C_m$",
  "title_text": "On lifting representations and actions on curves of the metacyclic groups C_p^srtimes C_m",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/3dxm448exafgj",
  "venue": "arXiv",
  "versions": []
}