Arboreal Man

Arboreal Galois Representations

Let GQ be the absolute Galois group of Q, and let T be the complete rooted d-ary tree, where d ≥ 2. In this article we study “arboreal” representations of GQ 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 GQ. We first give some methods of constructing arboreal rep...

Local Arboreal Representations

Let K be a field complete with respect to a discrete valuation v of residue characteristic p. Let f(z) ∈ K[z] be a separable polynomial of the form z − c. Given a ∈ K, we examine the Galois groups and ramification groups of the extensions of K generated by the solutions to f(z) = a. The behavior depends upon v(c), and we find that it shifts dramatically as v(c) crosses a certain value: 0 in the...

Stability During Arboreal Locomotion

Arboreal spiders in eastern hemlock.

Eastern hemlock [Tsuga canadensis (L.) Carrière] is a foundation species in forests of eastern North America that plays a key role in ecosystem function. It is highly susceptible to the exotic invasive hemlock woolly adelgid (Adelges tsugae Annand), which is causing widespread hemlock mortality. We surveyed the spider communities of eastern hemlock and deciduous canopies over 2 yr, collecting ...

The Image of an Arboreal Galois Representation

Much is known regarding images of p-adic Galois representations coming from subquotients of étale cohomology groups of varieties over number fields. In particular, the Mumford-Tate conjecture gives them up to subgroups of finite index, and has been proved in many cases by Serre. In the analogous situation of arboreal Galois representations little is known. In this paper we give a conjectural de...