Ramified Approximation and Semistable Reduction
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913652801536000 |
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| author | Faber, Xander |
| author_facet | Faber, Xander |
| contents | Let $K$ be a complete discretely valued field. An extension $L/K$ is "weakly totally ramified" if the residue extension is purely inseparable. We sharpen a result of Ax by showing that any Galois-invariant disk in the algebraic closure of $K$ contains an element that generates a separable weakly totally ramified extension. As an application, we prove that elliptic curves and dynamical systems on $\mathbb{P}^1$ achieve semistable reduction over a separable weakly totally ramified extension of the base field. We also obtain several arithmetic consequences for torsion points on elliptic curves and preperiodic points for dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramified Approximation and Semistable Reduction Faber, Xander Number Theory Let $K$ be a complete discretely valued field. An extension $L/K$ is "weakly totally ramified" if the residue extension is purely inseparable. We sharpen a result of Ax by showing that any Galois-invariant disk in the algebraic closure of $K$ contains an element that generates a separable weakly totally ramified extension. As an application, we prove that elliptic curves and dynamical systems on $\mathbb{P}^1$ achieve semistable reduction over a separable weakly totally ramified extension of the base field. We also obtain several arithmetic consequences for torsion points on elliptic curves and preperiodic points for dynamical systems. |
| title | Ramified Approximation and Semistable Reduction |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.12089 |