Last update: 01/27/08
Abstract
This book is a revised and expanded version of the lecture notes from the Berkovich Space seminar held at the University of Georgia during Spring, 2004. (Those notes are still available below for downloading.) In addition to providing a concrete introduction to Berkovich's theory of analytic spaces, we develop the foundations of potential theory on the Berkovich projective line. This has applications in arithmetic geometry to the study canonical heights, arithmetic intersection theory, and arithmetic dynamics, among other things. After describing in detail the topological structure of the Berkovich projective line, we introduce the Hsia kernel, the fundamental kernel for potential theory, and define a Laplacian operator on the Berkovich projective line. We then develop a theory of capacities, harmonic, and subharmonic functions, all of which are strikingly similar to their classical complex counterparts. Finally, we give some applications to non-archimedean dynamics, including the construction of a canonical probability measure on the Berkovich projective line attached to a rational function of degree at least 2.This book is not yet completely finished, though it is finally nearing completion. Updates will be posted periodically to this website. Comments are certainly welcome!
Analysis and Dynamics on the Berkovich Projective Line (with Robert Rumely)
Last update: 07/25/04
Abstract
This is a set of expanded lecture notes from the Berkovich Space seminar held at the University of Georgia during Spring, 2004.