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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.

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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