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 $d\ge 2$, and let $\beta \in {\mathbb {P}}^1(\overline{K})$. For all $n\in \mathbb {N}\cup \{\infty \}$, the Galois groups $G_n(\beta )={{\mathrm{Gal}}}(K(f^{-n}(\beta ))/K(\beta ))$ embed into ${{\mathrm{Aut}}}(T_n)$, 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 $[{{\mathrm{Aut}}}(T_\infty ):G_\infty (\beta )]<\infty$. 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.