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

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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.

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Notes

  1. Classical ramification, i.e. the ramification with support in rational points; classically ramified points are also called critical points, as in [16].

  2. Since \(\frac{dS}{dT}\) is invertible, we can put it in the form \(\frac{dS}{dT}=\epsilon T^{\sigma }(1+h(T))\), where for each \(\rho \in (r,1)\), \(|h(T)|_{\eta _{0,\rho }}<1\). Following the terminology of [29, Lemma 1.6] we say that \(\sigma \) is the order of \(\frac{dS}{dT}\).

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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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  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35131, Padua, Italy

    Velibor Bojković

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Bojković, V. Riemann–Hurwitz formula for finite morphisms of p-adic curves. Math. Z. 288, 1165–1193 (2018). https://doi.org/10.1007/s00209-017-1931-y

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