Time：13:30-15:55, May 21/28, June 4/ 11, 2026
Venue：E13-128
Course: An Introduction to Arithmetic Dynamics
Speaker：Paolo Dolce
Abstract:
This course offers a focused introduction to arithmetic dynamics, a field at the intersection of number theory and dynamical systems. The central theme is the study of algebraic and arithmetic properties of orbits of rational maps defined over global fields.
The course begins with the foundational concepts of discrete dynamical systems: orbits, periodic and preperiodic points, and the motivating finiteness questions that drive the subject, such as the structure of rational preperiodic points and integer points in orbits. We then develop height functions—essential tools from Diophantine geometry—providing a thorough treatment of absolute and logarithmic heights culminating in a complete proof of Northcott's theorem, which guarantees that only finitely many algebraic points of bounded degree and bounded height can be preperiodic. Finally, we construct the canonical height following Tate's method, a dynamically normalized height function that refines the naive height and encodes deep arithmetic information about the map. The course concludes with applications of these techniques to the uniform boundedness conjectures for rational preperiodic points, situating the material within current research frontiers.
