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Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1165–1193 | Cite as

Riemann–Hurwitz formula for finite morphisms of p-adic curves

  • Velibor Bojković
Article

Abstract

Given a finite morphism \(\varphi :Y\rightarrow X\) of quasi-smooth Berkovich curves over a complete, non-archimedean, nontrivially valued algebraically closed field k of characteristic 0, we prove a Riemann–Hurwitz formula relating their Euler–Poincaré characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are p-adic Runge’s theorem together with valuation polygons of analytic functions. Using the results obtained, we provide another point of view on Riemann–Hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic.

Keywords

Berkovich spaces Berkovich curves Riemann–Hurwitz formula p-Adic Runge’s theorem 

Notes

Acknowledgements

I would like to thank to my supervisor Francesco Baldassarri for proposing the problem of RH formula for finite morphisms of affinoid curves to me, and for his support during my work on this problem. I would also like to thank to my supervisor Denis Benois for his help during my stay in Bordeaux where the part of this work was done. I extend my thanks to Antoine Ducros, Marco Garuti, Elmar Große-Klönne, Jérôme Poineau and Michael Temkin for answering my many questions and especially to Jérôme and Antoine for their many suggestions on how to improve the present article. Finally I thank to the referee for many useful comments.

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly

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