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Finite index theorems for iterated Galois groups of cubic polynomials. (English) Zbl 07051737
Summary: 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.

MSC:
37P15 Dynamical systems over global ground fields
11G50 Heights
11R32 Galois theory
14G25 Global ground fields in algebraic geometry
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
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