  We prove that the mutual energy (or the intersection product in the sense of
Arakelov theory) of two dynamical systems associated to Latt\`es morphisms over
$\mathbf{\bar Q}$ is uniformly bounded below and deduce a proof of a conjecture
of Bogomolov-Fu-Tschinkel: the number of common images of torsion points of two
non-isomorphic elliptic curves over $\mathbf{C}$ by a standard morphism to the
projective line is uniformly bounded. The proof crucially relies on the theory
of Berkovich spaces over $\mathbf{Z}$ and on an original argument allowing to
obtain a global estimate from a central estimate (over a trivially valued
field).
