Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Étale fundamental groups of affinoid $p$-adic curves

Author: Mohamed Saïdi
Journal: J. Algebraic Geom. 27 (2018), 727-749
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 [Enhancements On Off] (What's this?)

  • 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
  • Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
  • Helmut P. Epp, Eliminating wild ramification, Invent. Math. 19 (1973), 235–249. MR 321929, DOI
  • 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, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. 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
  • 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

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.