Morphisms of Berkovich curves and the different function

Adina Cohen, Michael Temkin*, Dmitri Trushin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

Given a generically étale morphism f:Y→X of quasi-smooth Berkovich curves, we define a different function δf: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 δf is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δf 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 languageAmerican English
Pages (from-to)800-858
Number of pages59
JournalAdvances in Mathematics
Volume303
DOIs
StatePublished - 5 Nov 2016

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

  • Berkovich analytic spaces
  • The different
  • Topological ramification

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