Riemann–Hurwitz formula for finite morphisms of p-adic curves
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.
KeywordsBerkovich spaces Berkovich curves Riemann–Hurwitz formula p-Adic Runge’s theorem
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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