{
  "abstract": "Quadratic Chabauty computations for $X_0(N)^*$ are complicated by local height contributions at primes of bad reduction. For squarefree $N$, we explain a strategy for constructing a global $p$-adic height for which all the contributions at the primes of bad reduction vanish. As our first main result, we show that such a $p$-adic height exists for all squarefree levels $N>714$. In order to do this, we first give an explicit description of the minimal regular model of $X_0(N)^*$ at primes of bad reduction, and show this model is semi-stable. Each component of this model gives rise to a linear condition which ensures that the contribution at that component vanishes. Using embeddings of quadratic orders into quaternion algebras, we obtain an upper bound for the number of these conditions; when the genus of $X_0(N)^*$ is greater than one more than this bound, there is enough freedom to choose a suitable correspondence, simplifying quadratic Chabauty computations. As an application, we determine the rational points on several curves $X_0(N)^*$ for which this was not done before.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40207",
  "author_total": 3,
  "authors": [
    {
      "name": "Nikola Adžaga"
    },
    {
      "name": "Maarten Derickx"
    },
    {
      "name": "Timo Keller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9x0hbsa9fqt6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40207",
  "fulltext_url": null,
  "id": "9x0hbsa9fqt6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40207",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7215053334",
  "pdf_url": "https://arxiv.org/pdf/2609.40207",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9x0hbsa9fqt6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40207",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40207"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Semi-stable models, local heights and quadratic Chabauty for $X_0(N)^*$",
  "title_text": "Semi-stable models, local heights and quadratic Chabauty for X_0(N)^*",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/9x0hbsa9fqt6b",
  "venue": "arXiv",
  "versions": []
}