An alternative proof of the Faltings-Elkies bound

TL;DR

This paper presents a new proof of the Faltings-Elkies bound on the average Arakelov-Green function for points on a Riemann surface, with effective bounds related to local coordinate coverings.

Contribution

It provides an alternative, effective proof of the Faltings-Elkies bound, improving understanding of the asymptotic behavior of the Arakelov-Green function.

Findings

Bound on the average Arakelov-Green function grows like O((log n)/n)

Proof is effective with respect to local coordinate coverings

Offers a new perspective on the Faltings-Elkies bound

Abstract

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 $O((\log n)/n)$. Our result is effective in terms of bounds of the Arakelov-Green function with respect to a given covering by local coordinates.