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Organizing Committee
University of Hawaii
Mathematics Department
City University of New York (CUNY)
University of Rochester
University of Michigan
Abstract
This workshop will examine global arithmetic dynamics from the perspectives of number theory, algebraic geometry, and model theory. It will introduce aspects of this topic to a larger audience, and clarify connections between different perspectives. In addition, there will be extensive discussion periods in which participants can collaborate on theoretical and computational aspects of the topic.
Global Arithmetic: Image courtesy of M. Zieve
Problems
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.
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Tutorial Week Schedule
Monday, March 12, 2012
2:30 - 3:30 PM EDT
Fields with several automorphisms - Alice Medvedev, UC Berkeley
Location: 11th Floor Lecture Hall
3:30 - 4:00 PM EDT
Coffee/Tea Break
Location: 11th Floor Collaborative Space
4:00 - 5:00 PM EDT
The dynamics of birational maps over finite fields and their signatures - John A. G. Roberts, The University of New South Wales, Sydney, Australia
Location: 11th Floor Lecture Hall
Tuesday, March 13, 2012
10:30 - 11:50 AM EDT
Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University
Location: 11th Floor Lecture Hall
3:30 - 4:00 PM EDT
Coffee/Tea Break
Location: 11th Floor Collaborative Space
4:00 - 5:00 PM EDT
Professional Development: Ethics in research I
Location: 11th Floor Lecture Hall
Lecture Videos
Zoe Chatzidakis
Centre National de la Recherche Scientifique (CNRS)
March 22, 2012 • 12:00 AM
