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Ramified Approximation and Semistable Reduction

Xander Faber

TL;DR

Ramified Approximation sharpens Ax's result by showing that every $G_K$-invariant disk contains an element generating a separable, weakly totally wildly ramified extension, with controlled residue and value-group growth. The authors develop MacLane's approximants to construct such elements explicitly and then apply the construction to obtain semistable reduction for elliptic curves and dynamical systems on $\mathbb{P}^1$ over separable weakly totally ramified extensions, with precise degree bounds that depend on the residue characteristic and, in dynamics, on the degree $d$. The work delivers concrete algorithms and consequences for torsion structures and preperiodic points, and it clarifies the ramification data needed to achieve semistable models in 1-dimensional arithmetic geometry. Overall, the paper provides a unified, constructive framework to pass to semistable reduction via ramified extensions, with sharp bounds and broad applicability to both elliptic curves and dynamical systems.

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.

Ramified Approximation and Semistable Reduction

TL;DR

Ramified Approximation sharpens Ax's result by showing that every -invariant disk contains an element generating a separable, weakly totally wildly ramified extension, with controlled residue and value-group growth. The authors develop MacLane's approximants to construct such elements explicitly and then apply the construction to obtain semistable reduction for elliptic curves and dynamical systems on over separable weakly totally ramified extensions, with precise degree bounds that depend on the residue characteristic and, in dynamics, on the degree . The work delivers concrete algorithms and consequences for torsion structures and preperiodic points, and it clarifies the ramification data needed to achieve semistable models in 1-dimensional arithmetic geometry. Overall, the paper provides a unified, constructive framework to pass to semistable reduction via ramified extensions, with sharp bounds and broad applicability to both elliptic curves and dynamical systems.

Abstract

Let be a complete discretely valued field. An extension 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 contains an element that generates a separable weakly totally ramified extension. As an application, we prove that elliptic curves and dynamical systems on 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.
Paper Structure (12 sections, 28 theorems, 102 equations, 2 figures)

This paper contains 12 sections, 28 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on complete discretely valued fields
  3. The analytic affine line over K
  4. MacLane's method of approximants
  5. Diskoids and inductive valuations
  6. Local geometry at a type II point
  7. Ramified Approximation
  8. Elliptic Curves
  9. Residue characteristic different from 2 or 3
  10. Residue characteristic different from 2
  11. Residue characteristic 2
  12. Dynamical systems on P1

Key Result

Corollary 1.2

Let $f \in K[z]$ be a nonconstant polynomial of degree $d = qm$, where $q$ and $m$ are coprime, and where $q = 1$ if $\mathrm{char}(\tilde{K}) = 0$ and $q$ is a power of $\mathrm{char}(\tilde{K})$ otherwise. Assume that $D$ is a disk of finite radius containing the roots of $f$. There exists $\alpha

Figures (2)

  • Figure 1: An example showing the projection map locally collapsing $m(\zeta,\vec{w}) = 3$ branches at $\alpha \in \mathbf{A}^{1,\mathrm{an}}_{\mathbb{C}_K}$ to a single branch at $\zeta \in \mathbf{A}^{1,\mathrm{an}}_{K}$. Here $\operatorname{pr}^{-1}(\zeta)$ consists of two points.
  • Figure 2: A diagram of the positions of approximants for $f(z) = z^4 + 20z^2 + 292$ in $\mathbf{A}^{1,\mathrm{an}}_{\mathbb{Q}_2}$ and their pre-images in $\mathbf{A}^{1,\mathrm{an}}_{\mathbb{C}_2}$. Here $\zeta_i$ corresponds to the approximant $V_i$, while $\zeta_B$ corresponds to the infimum valuation on $B$. The minimum disk about the roots of $f$ is $D(z^2 + 2,3)$; the corresponding point downstairs, $\zeta_B$ is not an approximant. See Example \ref{['ex:not_approximant']}

Theorems & Definitions (66)

  • Remark 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Conjecture 1.6
  • Definition 2.1
  • Remark 2.2
  • ...and 56 more