In this paper, we compute $\omega_N^2$, the self-intersection of the dualizing sheaf (in the sense of Arakelov geometry) of the modular curve $X_0(N)$ for $N$ square-free coprime with 6. We show the equality $\omega_N^2=3g_N\log N(1+o(1))$ thus giving an explicit bound for the size of small points for this curve in the sense of Bogomolov's conjectures. To do this, we estimate several quantities attached to the Arakelov metric starting with Petersson's trace formula.
The dvi file has been removed, as this paper has now appeard in Inventiones
Mathematicae, vol 131, 1998, pages 645-674.