Canonical factorizations of morphisms of Berkovich curves

TL;DR

This paper establishes canonical factorizations of morphisms of Berkovich curves based on their metric properties, extending ramification theory and refining the Riemann-Hurwitz formula for these morphisms.

Contribution

It introduces a new canonical tower construction for finite morphisms of Berkovich curves, linking ramification theory with metric properties and harmonicity.

Findings

Existence of canonical local and global factorizations of Berkovich curve morphisms.

Refinement of the Riemann-Hurwitz formula for type 2 points.

Application of ramification group theory to Berkovich curve morphisms.

Abstract

We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the intermediate field extensions in the tower has a Herbrand function with only one break-point and there is at most one extension with trivial Herbrand function. We apply the result to the setting of finite morphisms of Berkovich curves where we prove the existence of canonical local and global factorization of such morphisms according to their metric properties. Finally, we use the canonical factorizations to prove harmonicity properties finite morphisms satisfy at each type 2 point: formulas that can be regarded as a refinement of the Riemann-Hurwitz formula for such morphisms.