{
  "abstract": "Motivated by Clark's work on Euclidean quadratic forms, we introduce trace Euclidean and house Euclidean quadratic lattices over totally real number fields, and relate these concepts through the $p$-norm Euclidean property. We prove that, as $F$ varies over all totally real number fields of arbitrary degree, there are only finitely many positive definite trace Euclidean $\\mathcal{O}_{F}$-lattices of arbitrary rank $n\\ge1$, up to isometry. For every $p\\in[1,\\infty]$, the same finiteness holds for positive definite $p$-norm Euclidean lattices. We also classify all positive definite trace Euclidean lattices of rank one over real quadratic fields: there are exactly six isometry classes, and every such lattice is free. As a corollary, the trace Euclidean real quadratic fields are precisely $\\mathbb{Q}(\\sqrt{2})$, $\\mathbb{Q}(\\sqrt{5})$, and $\\mathbb{Q}(\\sqrt{13})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34604",
  "author_total": 1,
  "authors": [
    {
      "name": "Zilong He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2zmjn89hh3h09.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34604",
  "fulltext_url": null,
  "id": "2zmjn89hh3h09",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34604",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223952990",
  "pdf_url": "https://arxiv.org/pdf/2610.34604",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2zmjn89hh3h09.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34604",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34604"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields",
  "title_text": "Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2zmjn89hh3h09",
  "venue": "arXiv",
  "versions": []
}