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Ramified Approximation and Semistable Reduction (2407.12089v2)
Published 16 Jul 2024 in math.NT
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.