The uniform boundedness conjecture in arithmetic dynamics

January 14 to January 18, 2008

at the

American Institute of Mathematics, Palo Alto, California

organized by

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

This workshop, sponsored by AIM and the NSF, 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 (p-adic) 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 one-dimensional rational maps, and for projective morphisms of higher dimension.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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