Galois representations from pre-image trees: an arboreal survey

Abstract

Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges according to the action of phi. The absolute Galois group of K acts on T by tree automorphisms, giving a subgroup G(phi) of the group Aut(T) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G(phi) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of phi. Following an overview of the history of the subject and two of its fundamental questions, we survey cases where G(phi) is known to have finite index in Aut(T). While it is tempting to conjecture that such behavior should hold in general, we exhibit four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G(phi), thanks in part to a surprising application of probability theory. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of phi, which we survey briefly. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of phi proves decisive, just as the topology of these orbits is decisive in complex dynamics.