Abstract : Berkovich spaces over $\mathbb{Z}$ may be seen as fibrations containing complex analytic spaces as well as $p$-adic analytic spaces, for every prime number $p$. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P1 of torsion points of two elliptic curves.

Keywords : Berkovich spaces over $\mathbb{Z}$; Lattès morphisms; elliptic curves; torsion points; Arakelov theory