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,
LiangChung 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:

Classical complex dynamics of (quadratic) polynomial maps and their
associated moduli spaces.

Nonarchimedean (padic) dynamics and dynamics on Berkovich spaces.

Dynamical modular curves and their Jacobians.

Equidistribution theory of Galois orbits of preperiodic points.

Global methods from arithmetic geometry used to describe
integral and rational points on varieties.
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 onedimensional 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: