## Abstract

We show that the metric structure of morphisms f:Y→X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ=(Γ_{Y},Γ_{X}) of f, the sets N_{f,≥n} of points of Y of multiplicity at least n in the fiber are radial around Γ_{Y} with the radius changing piecewise monomially along Γ_{Y}. In this case, for any interval l=[z,y]⊂Y connecting a point z of type 1 to the skeleton, the restriction f|_{l} gives rise to a profile piecewise monomial function φ_{y}:[0,1]→[0,1] that depends only on the type 2 point y∈Γ_{Y}. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φ_{y}} with y∈Γ_{Y} ^{(2)}. We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φ_{y} coincides with the Herbrand function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.

Original language | American English |
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Pages (from-to) | 438-472 |

Number of pages | 35 |

Journal | Advances in Mathematics |

Volume | 317 |

DOIs | |

State | Published - 7 Sep 2017 |

### Bibliographical note

Funding Information:This work was supported by the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement 268182.

Publisher Copyright:

© 2017 Elsevier Inc.

## Keywords

- Berkovich curves
- Herbrand function
- Wild ramification