On Arboreal Galois Representations of Rational Functions

TL;DR

This paper investigates the size and structure of arboreal Galois representations associated with rational functions, resolving key questions about their groups in special cases and advancing understanding of their arithmetic properties.

Contribution

It addresses open questions by Jones and Manes on the size of Galois groups for specific rational functions and their symmetries, providing new results and progress on conjectures.

Findings

Resolved a question about the size of G(, ) for periodic sequences.

Described Galois groups of iterates of certain quadratic polynomials.

Obtained a zero-density result for primes dividing sequence terms.

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 $C(\phi, \alpha)$.