{
  "abstract": "Let $p = 4q + 1$ be a prime integer, where $q$ is an odd prime. Let $L = \\mathbb{Q}(ζ_p)^+$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\\mathbb{Q}(ζ_p)$. In this paper, we prove that if $U_L^+=U_L^2$ (i.e., every totally positive unit is a square in $L$), then for any non-trivial subfield $K \\subseteq L$ and any non-trivial totally positive unit $\\varepsilon \\in U_K^+$, the absolute trace satisfies the sharp lower bound $\\Tr_{K/\\mathbb{Q}}(\\varepsilon) \\ge p - 4$. As an application, we show that a spherical fusion category $\\C$ with global dimension $\\dim(\\mathcal{C})=p=4q+1$ is pointed, where $2$ is a primitive root modulo $q$, or $p=4q+1<2^{22}$ and $p\\neq 29$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15643",
  "author_total": 2,
  "authors": [
    {
      "name": "Yue Meng"
    },
    {
      "name": "Zhiqiang Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e92k03t9wbc8d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15643",
  "fulltext_url": null,
  "id": "e92k03t9wbc8d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15643",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221768179",
  "pdf_url": "https://arxiv.org/pdf/2610.15643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e92k03t9wbc8d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A trace lower bound for totally positive units in $\\mathbb{Q}(ζ_p)^+$ and its application to fusion category",
  "title_text": "A trace lower bound for totally positive units in ℚ(ζ_p)^+ and its application to fusion category",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/e92k03t9wbc8d",
  "venue": "arXiv",
  "versions": []
}