The Galois theory of orbits in arithmetic dynamics

Original Announcement

The main topics for the workshop are:

• "Arboreal" Galois groups of fields generated by points in backward orbits of finite algebraic maps.
• "Dynatomic" Galois groups of fields generated by periodic points of finite algebraic maps.

Arboreal Galois groups sit naturally as subgroups of tree (or graph) automorphism groups, while dynatomic Galois groups are naturally subgroups of certain wreath products. A fundamental problem is to determine general conditions under which these dynamically generated Galois groups have finite index in the natural geometric groups that contain them. This is a dynamical analog of Serre's theorem on the size of the Galois groups generated by torsion points on elliptic curves. The goal of the workshop is to better understand these towers of Galois groups over number fields and over function field in both the one-dimensional and higher dimensional settings. For the latter, an initial goal is to give a geometric characterization of those maps for which one does not expect a finite index theorem to hold, analogous to the case of CM elliptic curves.

Material from the workshop

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop: