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Brauer–Manin and Descent Obstructions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2054)

Abstract

In Chap. 10 we have discussed obstructions to sections arising from arithmetic at p-adic places. Later in Chap. 16 we will discuss what is known about the local analogues of the section conjecture over real and p-adic local fields. The present Chapter concerns the usual next step, when the local problem is considered settled (which it is not for the section conjecture). In order to possibly arise from a common global rational point, the tuple of local solutions must survive known obstructions from arithmetic duality: the Brauer–Manin obstruction and the descent obstruction. We develop here the analogous obstructions to a collection of local sections against being the restriction of a common global section.The Brauer–Manin obstruction for adelic sections was developed in Stix ( J. Pure Appl. Algebra 215(6):1371–1397, 2011). The descent obstruction is actually the older sibling of the Brauer–Manin obstruction. The technique of descent goes back to Fermat. Descent using torsors under tori was developed and studied in detail by Colliot-Thélène and Sansuc, while later Harari and Skorobogatov ( Compos. Math. 130(3):241–273, 2002) analysed descent obstructions coming from non-abelian groups. Descent obstructions under finite groups have been thoroughly analysed by Stoll ( Algebra Number Theor. 1:349–391, 2007). The transfer of the descent obstruction to spaces of sections was essentially worked out in Harari and Stix ( Stix, J. (ed.), Contributions in Mathematical and Computational Science, vol. 2, 2012). Unlike a priori for adelic points, for adelic sections the constant finite descent obstruction turns out to be the only obstruction to globalisation, see Theorem 144. This gives a link between the section conjecture and strong approximation that either yields interesting applications to the section conjecture, see Corollary 158, or, if one is pessimistic, opens up an approach to disproving the section conjecture eventually.

Keywords

  • Fundamental Group
  • Rational Point
  • Diophantine Equation
  • Strong Approximation
  • Twisted Section

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.

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References

  1. Bourbaki, N.: Commutative algebra. Chapters 1–7, reprint. Elements of Mathematics, xxiv + 625 pp. Springer, New York (1998)

    Google Scholar 

  2. Bremner, A., Lewis, D.J., Morton, P.: Some varieties with points only in a field extension. Arch. Math. 43, 344–350 (1984)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Demarche, C.: Obstruction de descente et obstruction de Brauer–Manin étale. Algebra Number Theor. 3, 237–254 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Eriksson, D., Scharaschkin, V.: On the Brauer–Manin obstruction for zero-cycles on curves. Acta Arith. 135, 99–110 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Harari, D., Skorobogatov, A.N.: Non-abelian cohomology and rational points. Compos. Math. 130, 241–273 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Harari, D., Stix, J.: Descent obstruction and fundamental exact sequence. In: Stix, J. (ed.) The Arithmetic of Fundamental Groups, PIA 2010. Contributions in Mathematical and Computational Science, vol. 2, pp. 147–166. Springer, Berlin (2012)

    Google Scholar 

  7. Harari, D., Szamuely, T.: Arithmetic duality theorems for 1-motives. J. Reine Angew. Math. 578, 93–128 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Jordan, C.: Recherches sur les substitutions, J. Liouville 17, 351–367 (1872)

    Google Scholar 

  10. Kim, M.: Remark on fundamental groups and effective Diophantine methods for hyperbolic curves. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds.) Number Theory, Analysis and Geometry, in Memory of Serge Lang, pp. 355–368. Springer, Berlin (2012)

    CrossRef  Google Scholar 

  11. Lind, C.-E.: Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, 97 pp. University of Uppsala (1940)

    Google Scholar 

  12. Manin, Y.I.: Le groupe de Brauer-Grothendieck en géométrie diophantienne. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 401–411. Gauthier-Villars, Paris (1971)

    Google Scholar 

  13. Milne, J.S.: Arithmetic Duality Theorems, 2nd edn, viii + 339 pp. BookSurge, LLC, Charleston (2006). ISBN: 1-4196-4274-X

    Google Scholar 

  14. Neukirch, J.: Algebraische Zahlentheorie, xiii + 595 pp. Springer, Berlin (1992)

    Google Scholar 

  15. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 323, xvi + 825 pp. Springer, Berlin (2008)

    Google Scholar 

  16. Pál, A.: Diophantine decidability for curves and Grothendieck’s section conjecture. http://arxiv.org/abs/1001.4969v1 arXiv: 1001.4969v2 [math.NT] (January 2010)

  17. Poonen, B.: Heuristics for the Brauer-Manin obstruction for curves. Exp. Math. 15, 415–420 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Poonen, B.: Insufficiency of the Brauer-Manin obstruction applied to étale covers. Ann. Math. (2) 171, 2157–2169 (2010)

    Google Scholar 

  19. Poonen, B., Voloch, J.F.: The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields. Ann. Math. (2) 171, 511–532 (2010)

    Google Scholar 

  20. Reichardt, H.: Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184, 12–18 (1942)

    MathSciNet  Google Scholar 

  21. Rungtanapirom, N.: Godeaux–Serre Varieties with Prescribed Arithmetic Fundamental Group, vi + 42 pp. Diplomarbeit, Heidelberg (2011)

    Google Scholar 

  22. Scharaschkin, V.: The Brauer-Manin obstruction for curves, preprint. http://www.jmilne.org/math/Students/b.pdf www.jmilne.org/math/Students/b.pdf (December 1998)

  23. Skorobogatov, A.: Beyond the Manin obstruction. Invent. Math. 135, 399–424 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Skorobogatov, A.: Torsors and rational points. In: Cambridge Tracts in Mathematics, vol. 144, viii + 187 pp. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  25. Skorobogatov, A.N.: Descent obstruction is equivalent to étale Brauer–Manin obstruction. Math. Ann. 344, 501–510 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Stix, J.: Projective anabelian curves in positive characteristic and descent theory for log étale covers, Thesis, Bonner Mathematische Schriften 354, xviii+118 (2002)

    Google Scholar 

  27. Stix, J.: The Brauer–Manin obstruction for sections of the fundamental group. J. Pure Appl. Algebra 215, 1371–1397 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Stoll, M.: Finite descent obstructions and rational points on curves. Algebra Numb. Theor. 1, 349–391 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Wittenberg, O.: Une remarque sur les courbes de Reichardt–Lind et de Schinzel. In: Stix, J. (ed.) The Arithmetic of Fundamental Groups. PIA 2010. Contributions in Mathematical and Computational Science, vol. 2, pp. 329–337. Springer, Berlin (2012)

    Google Scholar 

  30. Wolfrath, S.: Die scheingeometrische étale Fundamentalgruppe. Thesis, 62 pp., Universität Regensburg, preprint no. 03/2009

    Google Scholar 

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Stix, J. (2013). Brauer–Manin and Descent Obstructions. In: Rational Points and Arithmetic of Fundamental Groups. Lecture Notes in Mathematics, vol 2054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30674-7_11

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