An alternative proof of the Faltings-Elkies bound

We give an alternative proof of the Faltings-Elkies bound on the average value of the Arakelov-Green function in pairs of a given set of $n$ points on a Riemann surface, which grows asymptotically like $O((\log n)\mathrm{/}n)$. Our result is effective in terms of bounds of the Arakelov-Green function with respect to a given covering by local coordinates.