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Mathematische Annalen

, Volume 373, Issue 1–2, pp 37–72 | Cite as

Finite index theorems for iterated Galois groups of cubic polynomials

  • Andrew BridyEmail author
  • Thomas J. Tucker
Article
  • 225 Downloads

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.

Mathematics Subject Classification

Primary 37P15 Secondary 11G50 11R32 14G25 37P05 37P30 

Notes

Acknowledgements

We would like to thank Thomas Gauthier, Dragos Ghioca, Keping Huang, Rafe Jones, Nicole Looper, and Khoa Nguyen for many helpful conversations. We would also like to thank the referee for many valuable comments and corrections.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of MathematicsUniversity of RochesterRochesterUSA

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