Math 295
Arithmetic Dynamics
MW 1:30-2:45pm
January 25 - April 28, 2021

Lecture notes are available on request; contact me (Laura DeMarco) for more information.

The goal of the semester is to study a collection of Arithmetic Equidistribution Theorems addressing the geometry of points with "small height" in a projective algebraic variety (defined over a number field). The case of heights on P^1, especially those arising from dynamical systems, will play a key role. We will study topics such as:

- basics of complex dynamical systems for maps on P^1
- potential theory in C
- Berkovich spaces in dimension 1
- basics of p-adic dynamics
- potential theory on the Berkovich affine line
- height functions, adelic measures, and small points on P^1

and discuss the equidistribution theorems of many authors (Baker, Bilu, Chambert-Loir, Favre, Rivera-Letelier, Rumely, Thuillier, and Yuan), with ideas that go back to an important article of Szpiro-Ullmo-Zhang (1997) about abelian varieties.

Prerequisites. Standard graduate-level courses in Analysis, Algebra, and Geometry. Background in dynamical systems and algebraic number theory is helpful but not required.

Course details. The course will run synchronously, by Zoom, on Mondays and Wednesdays at 1:30pm Eastern time. Attendance and participation are important to me, and questions are always welcome. Learning does not happen passively; all are encouraged to work on the suggested Exercises. If you will take this course for a grade, then you will need to submit solutions to (many of) these Exercises. Please schedule individual meetings with me during the first two weeks of the semester so that I can know more about you, your background, and your goals.

Useful references

Complex textbooks:
John Milnor, Dynamics in One Complex Variable, Third Edition
Thomas Ransford, Potential Theory in the Complex Plane

Arithmetic textbooks:
Joseph Silverman, The Arithmetic of Dynamical Systems
Robert Benedetto, Dynamics in One Non-Archimedean Variable
Matthew Baker and Robert Rumely, Potential Theory and Dynamics on the Berkovich Projective Line
Enrico Bombieri and Walter Gubler, Heights in Diophantine Geometry

Articles:
Bilu 1997
Rumely 1999
Northcott 1950
Rumely Chapter 6 excerpt from Capacity Theory on Algebraic Curves
Brolin 1965
Hubbard-Papadopol 1994
Milnor on Lattes maps 2004
Hrushovski-Loeser-Poonen on Berkovich spaces 2014