Original Paper

Geometriae Dedicata

, Volume 124, Issue 1, pp 27-35

First online:

Arboreal Galois representations

  • Nigel BostonAffiliated withUniversity of Wisconsin
  • , Rafe JonesAffiliated withUniversity of Wisconsin Email author 

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Abstract

Let \(G_{\mathbb{Q}}\) be the absolute Galois group of \(\mathbb{Q}\), and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of \(G_{\mathbb{Q}}\) into the automorphism group of T, particularly in the case d =  2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of \(G_\mathbb{Q}\). We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of \(G_{\mathbb{Q}}\). Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of \(G_{\mathbb{Q}}\).

Keywords

Galois representation Rooted tree Tree automorphisms Pro-p group Iterates Monodromy groups

Mathematics Subjects Classifications

11F80 11R32 20E08 20E18