Descent Obstruction and Fundamental Exact Sequence

  • David Harari
  • Jakob Stix
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)


We establish a link between the descent obstruction against rational points and sections of the fundamental group extension that has applications to the Brauer–Manin obstruction and to the birational case of the section conjecture in anabelian geometry.


Exact Sequence Conjugacy Class Elliptic Curf Abelian Variety Group Scheme 
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  1. Bou98.
    N. Bourbaki. General topology. Chapters 1–4. Elements of Mathematics. Springer, 1998. Translated from the French, reprint of the 1989 English translation, vii+437 pp.Google Scholar
  2. CTS77.
    J.-L. Colliot-Thélène and J.-J. Sansuc. La descente sur une variété rationnelle définie sur un corps de nombres. C. R. Acad. Sci. Paris, 284:1215–1218, 1977.zbMATHGoogle Scholar
  3. Dem09.
    C. Demarche. Obstruction de descente et obstruction de Brauer-Manin étale. Algebra and Number Theory, 3(2):237–254, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  4. Duk95.
    W. Duke. The critical order of vanishing of automorphic L-functions with large level. Invent. Math., 119(1):165–174, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  5. EH08.
    H. Esnault and Ph. H. Hai. Packets in Grothendieck’s Section Conjecture. Adv. Math., 218(2):395–416, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  6. EW10.
    H. Esnault and O. Wittenberg. On abelian birational sections. Journal of the American Mathematical Society, 23:713–724, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  7. Fal83.
    G. Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., 73(3):349–366, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 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
  9. Har00.
    D. Harari. Weak approximation and non-abelian fundamental groups. Ann. Sci. école Norm. Sup. (4), 33(4):467–484, 2000.Google Scholar
  10. HS02.
    D. Harari and A. N. Skorobogatov. Non-abelian cohomology and rational points. Compositio Math., 130(3):241–273, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  11. HS09.
    D. Harari and T. Szamuely. Galois sections for abelianized fundamental groups. Math. Ann., 344(4):779–800, 2009. With an appendix by E. V. Flynn.Google Scholar
  12. KL91.
    V. A. Kolyvagin and D. Yu. Logachëv. Finiteness of X over totally real fields. Izv. Akad. Nauk SSSR Ser. Mat., 55(4):851–876, 1991. Russian, translation in Math. USSR-Izv. 39(1):829–853, 1992.Google Scholar
  13. Koe95.
    J. Koenigsmann. From p-rigid elements to valuations (with a Galois-characterization of p-adic fields). J. Reine Angew. Math., 465:165–182, 1995. With an appendix by Florian Pop.Google Scholar
  14. Koe05.
    J. Koenigsmann. On the “section conjecture” in anabelian geometry. J. Reine Angew. Math., 588:221–235, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  15. Maz78.
    B. Mazur. Modular curves and the Eisenstein ideal. IHES Publ. Math., 47:33–186, 1978.Google Scholar
  16. MR10.
    B. Mazur and K. Rubin. Ranks of twists of elliptic curves and Hilbert’s Tenth Problem. Invent. Math., 181(3):541–575, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Neu77.
    J. Neukirch. Über die absoluten Galoisgruppen algebraischer Zahlkörper. Journées Arithmétiques de Caen 1976, Astérisque, 41-42:67–79, 1977.Google Scholar
  18. Pop10.
    F. Pop. On the birational p-adic section conjecture. Compositio Math., 146(3):621–637, 2010.CrossRefzbMATHGoogle Scholar
  19. PS11.
    F. Pop and J. Stix. Arithmetic in the fundamental group of a p-adic curve: on the p-adic section conjecture for curves. Preprint, arXiv:math.AG/1111.1354, 2011.Google Scholar
  20. S + 08.
    W. A. Stein et al. Sage Mathematics Software (Version 3.1.4), The Sage Development Team, 2008,
  21. 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
  22. Ser68.
    J.-P. Serre. Corps locaux. Publications de l’Université de Nancago, No. VIII., Hermann, Paris, 1968. (deuxième édition).Google Scholar
  23. Ser94.
    J.-P. Serre. Cohomologie Galoisienne. Springer Verlag, 1994. (cinquième édition, révisée et complétée).Google Scholar
  24. Sko01.
    A. N. Skorobogatov. Torsors and rational points. Volume 144 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001.Google Scholar
  25. Sti08.
    J. Stix. On cuspidal sections of algebraic fundamental groups. Preprint, arXiv:math.AG/ 0809.0017v1. Philadelphia–Bonn, 2008.Google Scholar
  26. Sti10.
    J. Stix. On the period-index problem in light of the section conjecture. Amer. J. of Math., 132(1):157–180, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Sto06.
    M. Stoll. Finite descent obstructions and rational points on curves. Preprint, arXiv:math.NT/0606465v2. Draft version no. 8.Google Scholar
  28. Sto07.
    M. Stoll. Finite descent obstructions and rational points on curves. Algebra and Number Theory, 1(4):349–391, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  29. Sza09.
    T. Szamuely. Galois groups and fundamental groups, volume 117 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2009.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Université de Paris-Sud (Orsay)Orsay CedexFrance
  2. 2.MATCH,Universität HeidelbergHeidelbergGermany

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