Finite index theorems for iterated Galois groups of cubic polynomials

TL;DR

This paper establishes necessary and sufficient conditions for the Galois groups of iterated cubic polynomial preimages over certain fields to have finite index in automorphism groups of rooted trees, advancing understanding in arithmetic dynamics.

Contribution

It provides the first complete characterization of finite index Galois groups for iterated cubic polynomials over function fields, and conditional results over number fields.

Findings

Necessary and sufficient conditions for finite index over function fields.

Conditional proof for number fields assuming $abc$ and Vojta's conjectures.

Extension of finite index results to variants of the problem.

Abstract

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\text{Gal}(K(f^{-n}(\beta))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\text{Aut}(T_\infty):G_\infty]<\infty$. When $f$ is a cubic polynomial and $K$ is a function field of transcendence degree $1$ over an algebraic extension of $\mathbb{Q}$, we resolve this problem by proving a list of necessary and sufficient conditions for finite index. This is the first result that gives necessary and sufficient conditions for finite index, and can be seen as a dynamical analog of the Serre Open Image Theorem. When $K$ is a number field, our proof is conditional on both the $abc$ conjecture for $K$ and Vojta's conjecture for blowups of $\mathbb{P}^1\times\mathbb{P}^1$. We also use our approach to solve some natural variants of the finite index problem for modified trees.