The Galois theory of orbits in arithmetic dynamics

May 16 to May 20, 2016

at the

American Institute of Mathematics, San Jose, California

organized by

Rafe Jones, Michelle Manes, and Joseph Silverman

Original Announcement

This workshop will be devoted to the study of Galois properties of points in orbits of algebraic maps.

The main topics for the workshop are:

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

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Current Trends and Open Problems in Arithmetic Dynamics
by  Robert Benedetto, Laura DeMarco, Patrick Ingram, Rafe Jones, Michelle Manes, Joseph H. Silverman, Thomas J. Tucker
Local arboreal representations
by  Jacqueline Anderson, Spencer Hamblen, Bjorn Poonen, and Laura Walton
A large arboreal Galois representation for a cubic postcritically finite polynomial
by  Robert L. Benedetto, Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku,  Res. Number Theory 3 (2017), Art. 29, 21 pp.  MR3736808
Reduction of dynamic curves
by  John R. Doyle, Holly Krieger, Andrew Obus, Rachel Pries, Simon Rubinstein-Salzedo, and Lloyd W. West
Dynamical Galois groups of trinomials and Odeon's conjecture
by  Nicole R. Looper
Galois groups of iterates of some unicritical polynomials
by  Michael R. Bush, Wade Hindes and Nicole R. Looper