{
  "abstract": "For an integer $b\\neq0,1$ let $K_b$ be the cubic field generated by the unique real root of $x^3-3bx-b^3$, let $η_0$ be its fundamental unit, and let $η_b$ be the explicit unit studied in the companion paper (arXiv:2609.37389). Write $η_b=η_0^{n(b)}$. Deciding whether $n(b)$ is divisible by $q$ amounts to finding the integral points of a plane curve $C_q$ of degree $2q$. We prove that for every integer $q\\ge1$ the curve $C_q$ is irreducible with $g(C_q)=\\binom{q-1}{2}$, the genus of a smooth plane curve of degree $q$ rather than of degree $2q$. The proof is not a genus computation but a structural one: $C_q$ is the inverse image, under the $q$-power map of a two-dimensional torus, of the curve traced by the minimal polynomial of $η_b$ as $b$ varies, and the formula falls out of two applications of Riemann-Hurwitz in the resulting tower. As a consequence the problem splits into exactly three regimes: $q=2$ rational, $q=3$ elliptic, and $q\\ge4$ of general type. In particular only finitely many $b$ have $q\\mid n(b)$ for each $q\\ge4$. A lower bound for linear forms in two logarithms gives $n(b)<6082138$ for every $b$, so $n(b)\\in\\{1,2\\}$ for all but finitely many $b$, though not effectively. We add two computations: $n(b)$ is determined outright for $|b|\\le10^{30}$, and the rational points of $C_4$ are determined by quadratic Chabauty, giving $4\\mid n(b)\\iff b\\in\\{-75,-3,5,96\\}$. We state precisely which parts of the latter rest on certified arithmetic and which do not. Finally, we show that $K_b\\cong K_{b'}$ with $b\\neq b'$ holds for $\\{b,b'\\}=\\{-1,5\\}$, and otherwise only if one of the two parameters, $c$ say, satisfies $|c|>10^{30}$ and $n(c)$ is divisible by a prime $q\\ge5$. There are only finitely many such pairs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07333",
  "author_total": 2,
  "authors": [
    {
      "name": "Iwao Kimura"
    },
    {
      "name": "Hikaru Umemoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jdykkk9rpx6t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07333",
  "fulltext_url": null,
  "id": "7jdykkk9rpx6t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07333",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07333",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jdykkk9rpx6t.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07333",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07333"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units",
  "title_text": "On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/7jdykkk9rpx6t",
  "venue": "arXiv",
  "versions": []
}