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.