## Abstract

Given a generically étale morphism f:Y→X of quasi-smooth Berkovich curves, we define a different function δ_{f}:Y→[0,1] that measures the wildness of the topological ramification locus of f. This provides a new invariant for studying f, which cannot be obtained by the usual reduction techniques. We prove that δ_{f} is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δ_{f} can be used to explicitly construct the simultaneous skeletons of X and Y. As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p.

Original language | American English |
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Pages (from-to) | 800-858 |

Number of pages | 59 |

Journal | Advances in Mathematics |

Volume | 303 |

DOIs | |

State | Published - 5 Nov 2016 |

### Bibliographical note

Funding Information:This work was supported by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement 268182 and BSF grant 2010255 .

Publisher Copyright:

© 2016 Elsevier Inc.

## Keywords

- Berkovich analytic spaces
- The different
- Topological ramification