The uniform boundedness conjecture in arithmetic dynamics

January 14 to January 18, 2008

at the

American Institute of Mathematics, San Jose, California

organized by

Matthew Baker, Robert Benedetto, Liang-Chung Hsia, and Joseph H. Silverman

Original Announcement

This workshop will be devoted to arithmetic properties of preperiodic points for morphisms on projective space. It is known that such morphisms have only finitely many preperiodic points defined over any given number field. A fundamental conjecture in arithmetic dynamics asserts that there is a uniform bound for the number of such points that depends only on the degree of the field, the degree of the map, and the dimension of the space. This is a dynamical analog of the conjecture that torsion on abelian varieties is uniformly bounded by the degree of the field and the dimension of the variety.

A primary goal of the workshop is to develop tools and a strategy for proving the first (highly) nontrivial case of the uniform boundedness conjecture in dynamics, namely for quadratic polynomials in one variable over Q. This special case represents a dynamical analog of Mazur's theorem that elliptic curves over Q have bounded torsion. Among the areas that may prove useful in attacking the uniform boundedness conjecture are:

The goal is to bring together experts in these diverse areas and have them combine their knowledge to create new approaches to the study of arithmetic properties of periodic and preperiodic points for (quadratic) polynomials, for one-dimensional rational maps, and for projective morphisms of higher dimension.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Preperiodic points and unlikely intersections
by  Matthew Baker and Laura DeMarco
On the number of rational iterated pre-images of the origin under quadratic dynamical systems
by  Xander Faber and Benjamin Hutz
Uniform bounds on pre-images under quadratic dynamical systems
by  Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, and Michael E. Zieve
Rational 6-cycles under iteration of quadratic polynomials
by  Michael Stoll