#
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 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

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. * MR3736808Reduction 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