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arXiv · 2407.12089

Ramified Approximation and Semistable Reduction

Xander Faber
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Abstract

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.

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Xander Faber. 2025-01-16. Ramified Approximation and Semistable Reduction. https://arxiv.org/abs/2407.12089

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