An Analytic Description of Local Intersection Numbers at Non-Archimedian   Places for Products of Semi-Stable Curves
  We generalise a formula of Shou-Wu Zhang, which describes local arithmetic intersection numbers of three Cartier divisors with support in the special fibre on a a self-product of a semi-stable arithmetic surface using elementary analysis. By an approximation argument, Zhang extends his formula to a formula for local arithmetic intersection numbers of three adelic metrized line bundles on the self-product of a curve with trivial underlying line bundle. Using the results on intersection theory from arXiv:1404.1623 [math.AG] we generalize these results to d-fold self-products for arbitrary d. For the approximations to converge, we have to assume that d satisfies the vanishing condition 4.7 from arXiv:1404.1623 [math.AG], which is true at least for $d\in \{2,3,4,5\}$. 
Johannes Kolb
math.AG