Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, RI
The aim of this workshop is to bring together leading researchers in global arithmetic dynamics and related fields to discuss recent result. In particular, we hope to attract researchers who work in arithmetic geometry, algebraic geometry, model theory, and computational algebra and number theory, with the dual goals of introducing the field of arithmetic dynamics and encouraging interactions among people working in these varied fields.
Problem 1: The Uniform Boundedness Conjecture. This fundamental conjecture in arithmetic dynamics says that for given positive integers D, n, and d with d>1, if K/Q is a number field of degree D and if f:Pn→Pn is a morphism of degree d defined over K, then the number of K-rational preperiodic points of f is bounded by a constant depending only on D, n, and d. It is a vast generalization of the Mazur-Merel theorem on uniform boundedness of torsion points on elliptic curves.
Problem 2: Dynamical Intersection Theorems. Two fundamental arithmetic results for abelian varieties are theorems of Raynaud and Faltings, orginally formulated as conjectures by Manin-Mumford and Mordell-Lang, respectively. There are a number of dynamical analogues of these conjectures, which roughly say that the orbit of a point should intersect a subvariety only finitely often unless the orbit of the entire subvariety has special properties. Only special cases of the dynamical conjectures have been proven. We expect these problems to be a major focus of the workshop.
Problem 3: Global Applications of Equidistribution Theorems. Let (xi) be a of sequence of algebraic points whose f-canonical heights go to zero. Under suitable hypotheses, it is known that the Galois conjugates of the xi are equidistributed with respect to the complex and Berkovich invariant measures. A focus of the program will be on global arithmetic applications of this and other similar equidistribution theorems.
Problem 4: Arithmetic Dynamics Over Function Fields. Many theorems in arithmetic geometry were first proven in the easier setting of function fields. A theme for the workshop will be the analogy for arithmetic dynamics between number fields and function fields.
Problem 5: Local-Global Problems. There are many local-global principles in arithmetic geometry, such as those related to the Hasse and Brauer-Manin obstructions. The workshop will explore analogous local-global principles for dynamical systems, including especially the distribution of orbits modulo primes and/or in p-adic or complex neighborhoods.