Advertisement

Cuspidal Sections and Birational Analogues

  • Jakob Stix
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2054)

Abstract

As Grothendieck noticed in his letter to Faltings, a k-rational point in the boundary of a good compactification contributes a packet of sections, see Grothendieck ( Schneps, L., Lochak, P. (eds.), LMS Lecture Notes vol. 242, 1997), page 8, not necessarily accounted for by rational points of the variety. These sections will here be called cuspidal sections. In the general framework, cuspidal sections are best constructed by Deligne’s theory of tangential base points, see Deligne ( Mathematical Sciences Research Institute Publications, vol. 16, 1989). Cuspidal sections have been studied by Nakamura in the profinite setting (Nakamura, J. Reine Angew. Math. 405:117–130, 1990; J. Reine Angew. Math. 411:205–216, 1990; Math. Zeit. 206(4):617–622, 1991), while Esnault and Hai ( Adv. Math. 218(2):395–416, 2008) discussed the analogue in a tannakian setting.We will define packets of cuspidal sections and show in Proposition 250 that for hyperbolic curves over arithmetic base fields these packets are either empty or uncountable. Then we describe and extend Nakamura’s characterisation of cuspidal sections in terms of cyclotomically normalized pro-cyclic subgroups in order to cover the description foreseen by Grothendieck in his letter.We will also discuss the birational analogue of the section conjecture when smooth geometrically connected varieties are replaced by their function fields: by passing to the limit over all open subschemes, all sections in this birational setup are predicted to become cuspidal. For curves over finite extensions of \({\mathbb{Q}}_{p}\) the birational section conjecture holds by Koenigsmann’s Theorem, and we have even a 2-step nilpotent pro-p version due to Pop. Over an algebraic number field there are some partial results, see Theorem 265, and for curves over \(\mathbb{Q}\) we even have a full group theoretic description of all birationally liftable sections in Theorem 269 with respect to certain \({GL }_{2}\)-representations.For ease of exposition we will work in characteristic 0 only.

Keywords

Function Field Finite Extension Open Subschemes Algebraic Number Field Mathematical Science Research Institute 
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.

References

  1. Bo98.
    Bourbaki, N.: Commutative algebra. Chapters 1–7, reprint. Elements of Mathematics, xxiv + 625 pp. Springer, New York (1998)Google Scholar
  2. BoEm12.
    Borne, N., Emsalem, M.: Un critère d’épointage des sections \(\mathcal{l}\)-adiques, preprint. http://arxiv.org/abs/1201.4589v1 arXiv: 1201.4589v1 [math.NT] (January 2012)
  3. De89.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois groups over \(\mathbb{Q}\) (Berkeley, CA, 1987). Mathematical Sciences Research Institute Publications, vol. 16, pp. 79–297. Springer, Berlin (1989)Google Scholar
  4. EsHa08.
    Esnault, H., Hai, Ph.H.: Packets in Grothendieck’s section conjecture. Adv. Math. 218, 395–416 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. EsWi10.
    Esnault, H., Wittenberg, O.: On abelian birational sections in characteristic 0. J. Am. Math. Soc. 23, 713–724 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Gr83.
    Grothendieck, A.: Brief an Faltings (27/06/1983). In: Schneps, L., Lochak, P. (eds.) Geometric Galois Action 1. LMS Lecture Notes, vol. 242, pp. 49–58. Cambridge (1997)Google Scholar
  7. HaSx12.
    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
  8. Ho12.
    Hoshi, Y.: Conditional results on the birational section conjecture over small number fields, preprint, RIMS-1742 (February 2012)Google Scholar
  9. KaLa81.
    Katz, N., Lang, S.: Finiteness theorems in geometric classfield theory, with an appendix by K.A. Ribet. Enseign. Math. (2) 27, 285–319 (1981)Google Scholar
  10. Ko05.
    Koenigsmann, J.: On the ‘section conjecture’ in anabelian geometry. J. Reine Angew. Math. 588, 221–235 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Na90a.
    Nakamura, H.: Rigidity of the arithmetic fundamental group of a punctured projective line. J. Reine Angew. Math. 405, 117–130 (1990)MathSciNetzbMATHGoogle Scholar
  12. Na90b.
    Nakamura, H.: Galois rigidity of the étale fundamental groups of punctured projective lines. J. Reine Angew. Math. 411, 205–216 (1990)MathSciNetzbMATHGoogle Scholar
  13. Na91.
    Nakamura, H.: On Galois automorphisms of the fundamental group of the projective line minus three points. Math. Zeit. 206, 617–622 (1991)zbMATHCrossRefGoogle Scholar
  14. Po10.
    Pop, F.: On the birational p-adic section conjecture. Compos. Math. 146, 621–637 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Sa10.
    Saïdi, M.: Good sections of arithmetic fundamental groups. http://arxiv.org/abs/1010.1313v1arXiv: 1010.1313v1 [math.AG] (October 2010)
  16. Sa12.
    Saïdi, M.: Around the Grothendieck anabelian section conjecture. In: Coates, J., Kim, M., Pop, F., Saïdi, M., Schneider, P. (eds.) Non-abelian Fundamental Groups and Iwasawa Theory. Cambridge University Press, Cambridge (2012)Google Scholar
  17. Sx08.
    Stix, J.: On cuspidal sections of algebraic fundamental groups. In: Nakamura, H., Pop, F., Schneps, L., Tamagawa, A. (eds.) Galois-Teichmller Theory and Arithmetic Geometry. Proceedings for a Conferences in Kyoto (October 2010). Advanced Studies in Pure Mathematics, vol. 63, pp. 519–563. http://de.arxiv.org/abs/0809.0017v1arXiv: 0809.0017v1 [math.AG] (2012)
  18. Sx12a.
    Stix, J.: Birational p-adic Galois sections in higher dimensions. Israel J. Math. http://arxiv.org/abs/1202.2781arXiv: 1202.2781v1 [math.AG] (February 2012, to appear)
  19. Sx12b.
    Stix, J.: On the birational section conjecture with local conditions, preprint. http://arxiv.org/abs/1203.3236arXiv: 1203.3236v1 [math.AG] (March 2012)
  20. St06.
    Stoll, M.: Finite descent obstructions and rational points on curves, draft version no. 8. http://arxiv.org/pdf/math/0606465v2arXiv: 0606465v2 [math.NT] (November 2006)
  21. Wo09.
    Wolfrath, S.: Die scheingeometrische étale Fundamentalgruppe. Thesis, 62 pp., Universität Regensburg, preprint no. 03/2009Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jakob Stix
    • 1
  1. 1.Mathematics Center Heidelberg (MATCH)University of HeidelbergHeidelbergGermany

Personalised recommendations