Local fields, iterated extensions, and Julia Sets

TL;DR

The paper studies how the ramification of the backward-orbit fields $K_{\infty}=\bigcup_n K_n$ for the unicritical polynomial $f(z)=z^\ell-c$ over a local field is governed by the constant term valuation $v(c)$ and the Berkovich Julia set. It unifies dynamical and arithmetic tools by analyzing the Newton polygon of $F(z)=(z+y)^\ell-y^\ell-d$, classifying Berkovich Julia sets across regimes determined by $v(c)$ and two cutoff values $v_{\infty}$ and $\nu_{\mathrm{good}}$. The main results show that $v(c)<v_{\infty}$ yields a finite extension $K_{\infty}/K$, $v_{\infty}\le v(c)<0$ (and $v(c)\ge0$) yields infinite but wildly ramified extensions, while $v(c)\ge\nu_{\mathrm{good}}$ corresponds to potential good reduction. This dynamical perspective clarifies the connection between Berkovich Julia sets and ramification, and extends previous specialcases to all $\ell\ge2$.

Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$, and let $f(z) = z^\ell - c \in K[z]$ be a separable polynomial. We explore the connection between the valuation $v(c)$ and the Berkovich Julia set of $f$. Additionally, we examine the field extensions generated by the solutions to $f^n(z) = α$ for a root point $α\in K$, highlighting the interplay between the dynamics of $f$ and the ramification in the corresponding extensions.

Theorems & Definitions (43)

• Theorem
• Remark 1.1
• Definition 2.1
• Lemma 2.2: Ben01
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Lemma 3.1
• proof