2019 Nov 22, Jamie Juul, University of British Columbia, Arboreal Galois representations

Maseeh Mathematics + Statistics Colloquium

Jamie Juul, University of British Columbia
Arboreal Galois representations

The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding ℓ-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point α under iterates of a rational map f (that is, points that eventually map to α as we apply f repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation.