Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1165–1193

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

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.

## References

1. 1.
Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. 2(1), 1–67 (2015)
2. 2.
Baldassarri, F.: Continuity of the radius of convergence of differential equations on $$p$$-adic analytic curves. Inventiones Mathematicae 182(3), 513–584 (2010)
3. 3.
Baldassarri, F., Kedlaya, K.: Harmonic functions attached to meromorphic connections on non-Archimedean curves. In preparation (2015)Google Scholar
4. 4.
Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, Volume 33 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1990)Google Scholar
5. 5.
Berkovich, V.G.: Étale cohomology for non-archimedean analytic spaces. Publications Mathématiques de l’IHÉS 78(1), 5–161 (1993)
6. 6.
Bojković, V.: On some applications of $$p$$-adic index theorem. In preparationGoogle Scholar
7. 7.
Bosch, S.: A rigid analytic version of M. Artin’s theorem on analytic equations. Math. Ann. 255(3), 395–404 (1981)
8. 8.
Bosch, S., Lütkebohmert, W.: Stable reduction and uniformization of abelian varieties I. Math. Ann. 270(3), 349–379 (1985)
9. 9.
Cohen, A., Temkin, M., Trushin, D.: Morphisms of Berkovich curves and the different function. Adv. Math. 303, 800–858 (2016)
10. 10.
Coleman, R.F.: Torsion points on curves and $$p$$-adic abelian integrals. Ann. Math. (2) 121(1), 111–168 (1985)
11. 11.
Coleman, R.F.: Reciprocity laws on curves. Compositio Mathematica 72(2), 205–235 (1989)
12. 12.
Coleman, R.F.: Stable maps of curves. Documenta Mathematica Extra Vol. Kato:217–225 (2003)Google Scholar
13. 13.
Coleman, R.F., De Shalit, E.: $$p$$-Adic regulators on curves and special values of $$p$$-adic L-functions. Inventiones Mathematicae 93(2), 239–266 (1988)
14. 14.
De Jong, A.J.: Étale fundamental groups of non-Archimedean analytic spaces. Compositio Mathematica 97(1–2), 89–118 (1995)
15. 15.
Ducros, A.: La structure des courbes analytiques. Manuscript (2014). http://www.math.jussieu.fr/ducros
16. 16.
Faber, X.: Topology and geometry of the Berkovich ramification locus for rational functions. I. Manuscripta Mathematica 142(3–4), 439–474 (2013)
17. 17.
Fresnel, J., Matignon, M.: Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique. Annali di Matematica 145(1), 159–210 (1986)
18. 18.
Garuti, M.A.: Prolongement de revêtements galoisiens en géométrie rigide. Compositio Mathematica 104(3), 305–331 (1996)
19. 19.
Gerritzen, L., Van der Put, M.: Schottky Groups and Mumford Curves, Volume 817 of Lecture Notes in Mathematics. Springer, Berlin (1980)
20. 20.
Große-Klönne, E.: Rigid analytic spaces with overconvergent structure sheaf. Journal für die Reine und Angewandte Mathematik 519, 73–95 (2000)
21. 21.
Große-Klönne, E.: De Rham cohomology of rigid spaces. Math. Z. 247(2), 223–240 (2004)
22. 22.
Große-Klönne, E.: Remark on the Čech cohomology of a coherent sheaf on dagger spaces. A letter to Francesco Baldassarri (2013)Google Scholar
23. 23.
Hartshorne, R.: Algebraic Geometry, Volume 52 of Graduate Texts in Mathematics. Springer Science and Business Media, Berlin (1977)Google Scholar
24. 24.
Huber, R.: Swan representations associated with rigid analytic curves. Journal für die Reine und Angewandte Mathematik 537, 165–234 (2001)Google Scholar
25. 25.
Kato, K.: Vanishing cycles, ramification of valuations, and class field theory. Duke Math. J. 55(3), 629–659 (1987)
26. 26.
Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Inventiones Mathematicae 2(4), 256–273 (1967)
27. 27.
Liu, Q.: Ouverts analytiques d’une courbe algébrique en géométrie rigide. Annales de l’Institut Fourier 37(3), 39–64 (1987)
28. 28.
Liu, Q.: Algebraic Geometry and Arithmetic Curves, Volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2002)Google Scholar
29. 29.
Lütkebohmert, W.: Riemann’s existence problem for a $$p$$-adic field. Inventiones Mathematicae 111(1), 309–330 (1993)
30. 30.
Poineau, J., Pulita, A.: The convergence Newton polygon of a $$p$$-adic differential equation IV: local and global index theorems. arXiv preprint arXiv:1309.3940 (2013)
31. 31.
Raynaud, M.: Revêtements de la droite affine en caractéristique $$p>0$$ et conjecture d’Abhyankar. Inventiones Mathematicae 116(1), 425–462 (1994)
32. 32.
Van Der Put, M.: The class group of a one-dimensional affinoid space. Annales de l’Institut Fourier 30(4), 155–164 (1980)
33. 33.
Van der Put, M.: De Rham cohomology of affinoid spaces. Compositio Mathematica 73(2), 223–239 (1990)
34. 34.
Welliaveetil, J.: A Riemann–Hurwitz formula for skeleta in non-Archimedean geometry. arXiv preprint arXiv:1303.0164 (2013)