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

arXiv comment21 pages