Beijing Institute of Mathematical Sciences and Applications

Let $p$ be any place of the rational field $\mathbb{Q}$ (can be the infinite place) and denote by $\mathbb{Q}_p$ the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value. We then set $C_p$ to be the completion of an algebraic closure of $\mathbb{Q}_p$. When $p$ is the infinite place of $\mathbb{Q}$, the field $C_p$, which is then the usual complex plane, is topologically nice (Haussdorf, simply connected, locally compact, spherically complete). However, when $p$ is a finite place, that is, a rational prime, the field $C_p$ is topologically "awful" (it is Haussdorf and complete, but totally disconnected, not locally compact and not spherically complete).

In the first part of this course, we will see that the Berkovich affine line, defined as the set of semi-norms on the ring of polynomials $C_p[X]$, turns $C_p$ into a "nice" topological space. More precisely, we will show that it is Haussdorf, locally compact, uniquely path-connected (so, it's a tree) and contains $C_p$ as a dense subset. After studying the topological properties of this space, we will start the study of complex $p$-adic dynamics on the Berkovich projective line, which is the "perfect" non-archimedean analogue of the classical complex dynamics.