Ramified Approximation and Semistable Reduction

TL;DR

This paper refines understanding of ramification in valued fields, showing how certain extensions facilitate semistable reduction of elliptic curves and dynamical systems, with implications for torsion and preperiodic points.

Contribution

It improves a result of Ax by demonstrating the existence of specific separable weakly totally ramified extensions within Galois-invariant disks.

Findings

Elliptic curves achieve semistable reduction over separable weakly totally ramified extensions.

Dynamical systems on projective line attain semistable reduction over such extensions.

New arithmetic consequences for torsion points and preperiodic points are established.

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.