Hilbert-Samuel Property

This chapter is devoted to the proof of the arithmetic Hilbert-Samuel formula. The strategy is to combine a local version of the Hilbert-Samuel formula with an argument of casting to Arakelov geometry over a trivially valued field. In fact, by the dominated convergence theorem, we can deduce from the local Hilbert-Samuel formula that the arithmetic Hilbert-Samuel formula holds up to a constant. Then, by using the Harder-Narasimhan R-filtration developed in and the interpretation of R-filtrations as norms over a trivially valued field, one can reduce the problem to an arithmetic Hilbert-Samuel formula over a trivially valued field, where the local geometry gives automatically the global statement.