Local arboreal representations

TL;DR

This paper investigates the Galois groups and ramification structures of iterated polynomial extensions over local fields, revealing how these properties change dramatically depending on valuation thresholds related to the polynomial's parameters.

Contribution

It provides a detailed analysis of how the Galois and ramification groups vary with valuation in local fields for polynomials of the form z^ell - c, highlighting critical valuation thresholds.

Findings

Behavior shifts at valuation zero when p does not divide ell.

Behavior shifts at valuation -p/(p-1) when p equals ell.

Galois group structures depend on the valuation of c.

Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p \nmid \ell$, and $-p/(p-1)$ in the case $p=\ell$.