On Arboreal Galois Representations of Rational Functions

Abstract

The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a global field $K$ on a tree $T(ϕ, α)$ of iterated preimages of $α\in \mathbb{P}^1(K)$ under $ϕ\in K(x)$ with $\text{deg}(ϕ) \geq 2$ induces a homomorphism $ρ: \text{Gal}(K^{\text{ksep}}/K) \to \text{Aut}(T(ϕ, α))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(ϕ,α) := \text{im} ρ= \underset{\leftarrow n}\lim\text{Gal}(K(ϕ^{-n}(α))/K)$. Specifically, we consider two cases for the pair $(ϕ, α)$: (1) $ϕ$ is such that the sequence $\{a_n\}$ defined by $a_0 = α$ and $a_n = ϕ(a_{n-1})$ is periodic, and (2) $ϕ$ commutes with a nontrivial Mobius transformation that fixes $α$. In the first case, we resolve a question posed by Jones about the size of $G(ϕ, α)$, and taking $K = \mathbb{Q}$, we describe the Galois groups of iterates of polynomials $ϕ\in \mathbb{Z}[x]$ that have the form $ϕ(x) = x^2 + kx$ or $ϕ(x) = x^2 - (k+1)x + k$. When $K = \mathbb{Q}$ and $ϕ\in \mathbb{Z}[x]$, arboreal Galois representations are a useful tool for studying the arithmetic dynamics of $ϕ$. In the case of $ϕ(x) = x^2 + kx$ for $k \in \mathbb{Z}$, we employ a result of Jones regarding the size of the group $G(ψ, 0)$, where $ψ(x) = x^2 - kx + k$, to obtain a zero-density result for primes dividing terms of the sequence $\{a_n\}$ defined by $a_0 \in \mathbb{Z}$ and $a_n = ϕ(a_{n-1})$. In the second case, we resolve a conjecture of Jones about the size of a certain subgroup $C(ϕ, α) \subset \text{Aut}(T(ϕ, α))$ that contains $G(ϕ, α)$, and we present progress toward the proof of a conjecture of Jones and Manes concerning the size of $G(ϕ, α)$ as a subgroup of $C(ϕ, α)$.