{
  "abstract": "Firstly, for branched covering maps $f\\colon Y\\to X$ between smooth, separated schemes of finite type over the integral ring $O_K$ of a non-archimedean local field $K$, the ramification divisor is found to coincide with the divisor of the Radon-Nikodym derivative of the Radon measure associated with an algebraic differential form on $Y$ against the pullback measure of one on $X$, both taking Borel sets of the space $Y(O_K)$ of $O_K$-rational points of $Y$ as input values. Secondly, the ultracontractivity of semigroups generated by Laplacian integral operators having a geodetic-distance-power kernel function is proven, using heat kernel estimates for ultrametric operators on $p$-adic balls. Thirdly, a series of $p$-adic Neumann Boundary Value Problems, depending on algebraic and pluricanonical differential forms with poles on spaces $X(O_K)$ coming from schemes is formulated and solved, extending previous work of the author. These results are then used to solve reconstruction problems on schemes: the divisor of a pluricanonical form with poles with at worst log-terminal singularities, as well as Weierstrass points of projective algebraic curves can be reconstructed via repeatedly finding weak solutions of $p$-adic Neumann Boundary Value Problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27751",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Erik Bradley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqqe188a4qpj0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27751",
  "fulltext_url": null,
  "id": "eqqe188a4qpj0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27751",
  "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"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214160546",
  "pdf_url": "https://arxiv.org/pdf/2609.27751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqqe188a4qpj0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.27751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Searching Schemes With $p$-Adic Neumann Boundary Value Problems",
  "title_text": "Searching Schemes With p-Adic Neumann Boundary Value Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/eqqe188a4qpj0",
  "venue": "arXiv",
  "versions": []
}