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PreprintCC BY 4.0arXiv

Semi-stable models, local heights and quadratic Chabauty for X0(N)*

  • Nikola Adžaga
  • Maarten Derickx
  • Timo Keller
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Abstract

Quadratic Chabauty computations for X0(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 X0(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 X0(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 X0(N)* for which this was not done before.

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Cite

BibTeXRIS
BibTeX
@misc{adzaga2026semistable,
  title = {{Semi-stable models, local heights and quadratic Chabauty for $X_0(N)^*$}},
  author = {Nikola Adžaga and Maarten Derickx and Timo Keller},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40207},
  eprint = {2609.40207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40207},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Semi-stable models, local heights and quadratic Chabauty for X_0(N)^*
AU  - Nikola Adžaga
AU  - Maarten Derickx
AU  - Timo Keller
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40207
UR  - https://arxiv.org/abs/2609.40207
AB  - 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.
ER  -