Finite index theorems for iterated Galois groups of cubic polynomials

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.