Étale fundamental groups of affinoid $p$-adic curves
Author:
Mohamed Saïdi
Journal:
J. Algebraic Geom. 27 (2018), 727-749
DOI:
https://doi.org/10.1090/jag/707
Published electronically:
June 29, 2018
MathSciNet review:
3846552
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove that the geometric étale fundamental group of a (geometrically connected) rigid smooth $p$-adic affinoid curve is a semi-direct factor of a certain profinite free group. We prove that the maximal pro-$p$ (resp. maximal prime-to-$p$) quotient of this geometric étale fundamental group is pro-$p$ free of infinite rank (resp. (pro-)prime-to-$p$ free of finite computable rank).
References
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- Marius van der Put, The class group of a one-dimensional affinoid space, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 4, 155–164. MR 599628
References
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math. 119 (1995), no. 2, 361–398. MR 1312505, DOI https://doi.org/10.1007/BF01245187
- Nicolas Bourbaki, Commutative algebra. Chapters 1–7, translated from the French, reprint of the 1972 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. MR 979760
- Helmut P. Epp, Eliminating wild ramification, Invent. Math. 19 (1973), 235–249. MR 0321929, DOI https://doi.org/10.1007/BF01390208
- Marco A. Garuti, Prolongement de revêtements galoisiens en géométrie rigide, Compositio Math. 104 (1996), no. 3, 305–331 (French, with English summary). MR 1424559
- Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck, Augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). MR 0354651
- M. Raynaud, Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425–462 (French). MR 1253200, DOI https://doi.org/10.1007/BF01231568
- Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132
- Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577
- Jean-Pierre Serre, Construction de revêtements étales de la droite affine en caractéristique $p$, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 341–346 (French, with English summary). MR 1071640
- Marius van der Put, The class group of a one-dimensional affinoid space, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 4, 155–164. MR 599628
Additional Information
Mohamed Saïdi
Affiliation:
College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom
Email:
m.saidi@exeter.ac.uk
Received by editor(s):
August 11, 2016
Received by editor(s) in revised form:
February 5, 2017, May 25, 2017, and June 7, 2017
Published electronically:
June 29, 2018
Dedicated:
In memory of Si M’hamed, my father
Article copyright:
© Copyright 2018
University Press, Inc.
