Berkovich spaces over $\mathbb{Z}$ and Schottky spaces

Berkovich spaces over $\mathbb{Z}$ look like fibrations that contain complex analytic spaces as well as $p$-adic analytic spaces for every prime number $p$. We will give a short introduction to those spaces and explain that they provide a convenient setting to parametrize certain natural families such as Schottky groups and Mumford curves over arbitrary local fields ($\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$, $\mathbb{F}_p((t))$, etc.). Moreover, the familiar action of a Schottky group on a open subset of the projective line carries over to this global setting, giving rise to a global uniformization morphism similar to that of Schottky and Mumford. This is joint work with Daniele Turchetti.