{
  "abstract": "Let $k$ be algebraically closed of characteristic $p$. Reichstein and Vistoli showed that every $\\Z/p^n$-torsor becomes at most one-dimensional over a field extension of degree prime to $p$, and Ledet conjectured $\\ed_k(\\Z/p^n)=n$. Using the finite-degree profile $\\edd d_k(τ)$ and its jump degrees introduced in \\cite{PA}, we study the generic $\\Z/p^n$-torsor $\\tgen$, whose profile records how many Artin--Schreier--Witt layers an extension of degree $\\le d$ can absorb. An Abel--Jacobi rigidity argument, in which the purely transcendental total space of $\\tgen$ forces divisor classes on twisted curves to be constant, shows that a $G$-stable linear system on a faithful $\\Z/p^n$-curve has degree $\\ge p^{n-1}$. Hence $\\edd d_k(\\tgen)\\ge2$ for $d<p^{n-1}$ and $d_1(\\tgen)=p^{n-1}$ for all $p$ and $n\\ge2$; in particular the truncation tower is optimal in degree $p^{n-2}$. We prove a quantitative Reichstein--Vistoli theorem, $d_1^{(p')}(\\tgen)\\le l_n(p)=1+(p-1)\\sum_{j=1}^{n-1}p^{2j-1}$, and show that it is attained: $d_1^{(p')}(\\tgen)=l_n(p)$ for all $p$ and $n$. We determine the profile at $(p,n)=(2,2)$ and every slot but the first at $(2,3)$, where the remaining slot is $\\ed_k(\\Z/8)\\in\\{2,3\\}$, that is, Ledet's conjecture at level $3$. We show that the subgroup formula $\\edd{p}_k(τ_n)=n-1$ is equivalent to Ledet's conjecture at level $n-1$ together with a statement about extensions of degree $\\le p$ disjoint from $L$, so that the profile relocates, but does not decide, Ledet's conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09298",
  "author_total": 1,
  "authors": [
    {
      "name": "Abhishek Shukla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/avb85kjrj0701.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09298",
  "fulltext_url": null,
  "id": "avb85kjrj0701",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09298",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221019357",
  "pdf_url": "https://arxiv.org/pdf/2610.09298",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/avb85kjrj0701.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.09298",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09298"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The finite-degree profile of essential dimension II: cyclic $p$-groups in characteristic $p$",
  "title_text": "The finite-degree profile of essential dimension II: cyclic p-groups in characteristic p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/avb85kjrj0701",
  "venue": "arXiv",
  "versions": []
}