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