{
  "abstract": "Let $K$ be a $p$-adic field. We define the notion of finite height for an étale $\\mathbb{Z}_p$-local system on a smooth affinoid adic space $\\mathcal{X}$ over $K$ with semistable reduction. Using analytic prismatic $F$-crystals and purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603), we prove that if an étale $\\mathbb{Z}_p$-local system over $\\mathcal{X}$ is of finite height then its pullback along a finite étale cover of $\\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.26043",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaustabh Mondal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpk4hfpegvw00.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.26043",
  "fulltext_url": null,
  "id": "cpk4hfpegvw00",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.26043",
  "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": "W7165912604",
  "pdf_url": "https://arxiv.org/pdf/2606.26043",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpk4hfpegvw00.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2606.26043",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.26043"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Potential semistability of Finite height Galois representations: Relative case",
  "title_text": "Potential semistability of Finite height Galois representations: Relative case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/cpk4hfpegvw00",
  "venue": "arXiv",
  "versions": []
}