Heidelberg Lectures on Fundamental Groups

Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)

Abstract

We survey topics related to étale fundamental groups, with emphasis on Grothendieck’s anabelian program, the Section Conjecture and Parshin’s proof of the geometric case of Mordell’s conjecture.

Keywords

Conjugacy Class Fundamental Group Abelian Variety Inverse Limit Smooth Projective Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bel79.
    G. V. Belyi. Galois extensions of a maximal cyclotomic field (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 43:267–276, 479, 1979.Google Scholar
  2. Bog91.
    F. Bogomolov. On two conjectures in birational algebraic geometry. In Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., pages 26–52. Springer, Tokyo, 1991.Google Scholar
  3. BT08.
    F. Bogomolov and Yu. Tschinkel. Reconstruction of function fields. Geom. Funct. Anal., 18:400–462, 2008.CrossRefMATHMathSciNetGoogle Scholar
  4. Col90.
    R. F. Coleman. Manin’s proof of the Mordell conjecture over function fields. Enseign. Math., 36:393–427, 1990.MATHMathSciNetGoogle Scholar
  5. Con06.
    B. Conrad. Chow’s K ∕ k-image and K ∕ k-trace, and the Lang-Néron theorem. Enseign. Math., 52:37–108, 2006.MATHMathSciNetGoogle Scholar
  6. CP09.
    S. Corry and F. Pop. The pro-p hom-form of the birational anabelian conjecture. J. reine angew. Math., 628:121–127, 2009.CrossRefMATHMathSciNetGoogle Scholar
  7. EH08.
    H. Esnault and Ph. H. Hai. Packets in Grothendieck’s section conjecture. Adv. Math., 218(2):395–416, 2008.CrossRefMATHMathSciNetGoogle Scholar
  8. Fal83.
    G. Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., 73:349–366, 1983.CrossRefMATHMathSciNetGoogle Scholar
  9. Fal98.
    G. Faltings. Curves and their fundamental groups (following Grothendieck, Tamagawa and Mochizuki), Séminaire Bourbaki, exposé 840. Astérisque, 252:131–150, 1998.MathSciNetGoogle Scholar
  10. Gil00.
    P. Gille. Le groupe fondamental sauvage d’une courbe affine en caractéristique p > 0. In J-B. Bost et al., editors, Courbes semi-stables et groupe fondamental en géométrie algébrique, volume 187 of Progress in Mathematics, pages 217–231. Birkhäuser, Basel, 2000.Google Scholar
  11. Gra65.
    H. Grauert. Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper. Publ. Math. IHES, 25:131–149, 1965.MathSciNetGoogle Scholar
  12. Gro83.
    A. Grothendieck. Brief an Faltings (27/06/1983). In L. Schneps and P. Lochak, editors, Geometric Galois Actions 1, volume 242 of LMS Lecture Notes, pages 49–58. Cambridge, 1997.Google Scholar
  13. HM09.
    Y. Hoshi and S. Mochizuki. On the combinatorial anabelian geometry of nodally nondegenerate outer representations. preprint RIMS-1677. 2009.Google Scholar
  14. Hos10.
    Y. Hoshi. Existence of nongeometric pro-p Galois sections of hyperbolic curves. Publ. Res. Inst. Math. Sci., 46(4):829–848, 2010.MATHMathSciNetGoogle Scholar
  15. HS12.
    D. Harari and J. Stix. Descent obstruction and fundamental exact sequence. In J. Stix, editor, The Arithmetic of Fundamental Groups - PIA 2010, volume 2 of Contributions in Mathematical and Computational Sciences. Heidelberg, 2012.Google Scholar
  16. Kah06.
    B. Kahn. Sur le groupe des classes d’un schéma arithmétique (avec un appendice de Marc Hindry). Bull. Soc. Math. France, 134:395–415, 2006.MATHMathSciNetGoogle Scholar
  17. Kah09.
    B. Kahn. Démonstration géométrique du théorème de Lang-Néron et formules de Shioda-Tate. In Motives and algebraic cycles: a celebration in honour of Spencer J. Bloch, volume 56, pages 149–155. Fields Institute Communications, Amer. Math. Soc., 2009.Google Scholar
  18. KL81.
    N. M. Katz and S. Lang. Finiteness theorems in geometric class field theory. L’Enseign. Math., 27:285–314, 1981.MathSciNetGoogle Scholar
  19. Kob76.
    S. Kobayashi. Intrinsic distances, measures and geometric function theory. Bull. Amer. Math. Soc., 82:357–416, 1976.CrossRefMATHMathSciNetGoogle Scholar
  20. Koe05.
    J. Koenigsmann. On the “section conjecture” in anabelian geometry. J. reine angew. Math., 588:221–235, 2005.CrossRefMATHMathSciNetGoogle Scholar
  21. Lan87.
    S. Lang. Introduction to complex hyperbolic spaces. Springer, Berlin, 1987.MATHGoogle Scholar
  22. Lan91.
    S. Lang. Number theory III, volume 60 of Encyclopaedia of Mathematical Sciences. Springer, Berlin, 1991.Google Scholar
  23. LS57.
    S. Lang and J-P. Serre. Sur les revêtements non ramifiés des variétés algébriques. Amer. J. Math., 79:319–330, 1957.Google Scholar
  24. Man63.
    Yu. I. Manin. Rational points on algebraic curves over function fields (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 27:1395–1440, 1963.MATHMathSciNetGoogle Scholar
  25. Mat55.
    A. Mattuck. Abelian varieties over p-adic ground fields. Ann. of Math., 62:92–119, 1955.CrossRefMATHMathSciNetGoogle Scholar
  26. Mat96.
    M. Matsumoto. Galois representations on profinite braid groups on curves. J. reine angew. Math., 474:169–219, 1996.CrossRefMathSciNetGoogle Scholar
  27. Moc99.
    S. Mochizuki. The local pro-p anabelian geometry of curves. Invent. Math., 138:319–423, 1999.CrossRefMATHMathSciNetGoogle Scholar
  28. Moc04.
    S. Mochizuki. The absolute anabelian geometry of hyperbolic curves, pages 77–122. Kluwer, Boston, 2004.Google Scholar
  29. Moc07.
    S. Mochizuki. Absolute anabelian cuspidalizations of proper hyperbolic curves. J. Math. Kyoto Univ., 47:451–539, 2007.MATHMathSciNetGoogle Scholar
  30. Neu77.
    J. Neukirch. Über die absoluten Galoisgruppen algebraischer Zahlkörper. In Journées Arithmétiques de Caen, Astérisque 41-42:67–79, 1977.Google Scholar
  31. NSW08.
    J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields. Volume 323 in Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 2nd edition, 2008.Google Scholar
  32. NTM01.
    H. Nakamura, A. Tamagawa, and S. Mochizuki. The Grothendieck conjecture on the fundamental groups of algebraic curves. Sugaku Expositions, 14:31–53, 2001.MathSciNetGoogle Scholar
  33. Par68.
    A. N. Parshin. Algebraic curves over function fields I (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 32:1191–1219, 1968.MATHMathSciNetGoogle Scholar
  34. Par90.
    A. N. Parshin. Finiteness theorems and hyperbolic manifolds. In P. Cartier et al., editors, The Grothendieck Festschrift, vol. III, volume 88 of Progress in Mathematics, pages 163–178. Birkhäuser, Boston, MA, 1990.Google Scholar
  35. Pop00.
    F. Pop. Alterations and birational anabelian geometry. In H. Hauser et al., editor, Resolution of singularities, volume 181 of Progress in Mathematics, pages 519–532. Birkhäuser, Basel, 2000.Google Scholar
  36. Pop10a.
    F. Pop. On the birational anabelian program initiated by Bogomolov I. Preprint. 2010. to appear in Inventiones MathematicaeGoogle Scholar
  37. Pop10b.
    F. Pop. On the birational p-adic section conjecture. Compositio Math., 146:621–637, 2010.CrossRefMATHGoogle Scholar
  38. Sch33.
    F. K. Schmidt. Mehrfach perfekte Körper. Math. Ann., 108:1–25, 1933.CrossRefMathSciNetGoogle Scholar
  39. SGA1.
    A. Grothendieck. Revêtements étale et groupe fondamental (SGA 1). Séminaire de géométrie algébrique du Bois Marie 1960-61, directed by A. Grothendieck, augmented by two papers by Mme M. Raynaud, Lecture Notes in Math. 224, Springer-Verlag, Berlin-New York, 1971. Updated and annotated new edition: Documents Mathématiques 3, Société Mathématique de France, Paris, 2003.Google Scholar
  40. Sha47.
    I. R. Shafarevich. On p-extensions (Russian). Mat. Sbornik, 20(62):351–363, 1947. English translation: Amer. Math. Soc. Translation Series 4 (1956), 59-72.Google Scholar
  41. ST09.
    M. Saïdi and A. Tamagawa. A prime-to-p version of Grothendieck’s anabelian conjecture for hyperbolic curves over finite fields of characteristic p > 0. Publ. Res. Inst. Math. Sci., 45:135–186, 2009.CrossRefMATHMathSciNetGoogle Scholar
  42. Sti02.
    J. Stix. Projective anabelian curves in positive characteristic and descent theory for log étale covers. Bonner Mathematische Schriften, 354, 2002.Google Scholar
  43. Sto07.
    M. Stoll. Finite descent obstructions and rational points on curves. Algebra and Number Theory, 1:349–391, 2007.CrossRefMATHMathSciNetGoogle Scholar
  44. Sza00.
    T. Szamuely. Le théorème de Tamagawa I. In J.-B. Bost et al., editors, Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), volume 187 of Progress in Mathematics, pages 185–201. Birkhäuser, Basel, 2000.Google Scholar
  45. Sza04.
    T. Szamuely. Groupes de Galois de corps de type fini [d’après Pop], Séminaire Bourbaki, exposé 923. Astérisque, 294:403–431, 2004.MathSciNetGoogle Scholar
  46. Sza09.
    T. Szamuely. Galois Groups and Fundamental Groups. Volume 117 of Cambridge Studies in Advanced Mathematics, University Press, 2009.Google Scholar
  47. Tam97.
    A. Tamagawa. The Grothendieck conjecture for affine curves. Compositio Math., 109:135–194, 1997.CrossRefMATHMathSciNetGoogle Scholar
  48. Uch77.
    K. Uchida. Isomorphisms of Galois groups of algebraic function fields. Ann. of Math., 106:589–598, 1977.CrossRefMATHMathSciNetGoogle Scholar
  49. Win84.
    K. Wingberg. Ein Analogon zur Fundamentalgruppe einer Riemannschen Fläche im Zahlkörperfall. Invent. Math., 77:557–584, 1984.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

Personalised recommendations