Advertisement

Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 545–564 | Cite as

On some anabelian properties of arithmetic curves

  • A. IvanovEmail author
Article
  • 91 Downloads

Abstract

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicated to describe the position of decomposition groups of points at the boundary of the scheme \({{\rm Spec}\, \mathcal{O}_{K, S}}\), where K is a number field and S a set of primes of K, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally to the fundamental group \({\pi_1({\rm Spec}\, \mathcal{O}_{K, S})}\) : the location of decomposition groups of boundary points inside it, the p-part of the cyclotomic character, the number of points on the boundary of all finite étale covers, etc. Under a certain finiteness hypothesis on Tate–Shafarevich groups with divisible coefficients, one can reconstruct all these quantities simply from the fundamental group.

Mathematics Subject Classification (2010)

11R34 11R37 14G32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andozhskii I.V.: Demushkin groups. Mat. Zametki 14(1), 121–126 (1973)MathSciNetGoogle Scholar
  2. 2.
    Chenevier G., Clozel L.: Corps de nombres peu ramifiés et formes automorphes autoduales. J. AMS 22(2), 467–519 (2009)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Demushkin S.P.: The group of the maximal p-extension of a local field. Izv. Akad. Nauk SSSR, Ser. Matem. 25, 326–346 (1961)Google Scholar
  4. 4.
    Ivanov, A.: Arithmetic and Anabelian Theorems for Stable Sets in Number Fields. Dissertation, Universität Heidelberg (2013)Google Scholar
  5. 5.
    Koch H.: Galois Theory of p-Extensions. 1st edn. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Neukirch J.: Kennzeichnung der p-adischen und der endlich algebraischen Zahlkörper. Invent. Math. 6, 296–314 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Neukirch, J.: Klassenkörpertheorie, Mannheim (1969)Google Scholar
  8. 8.
    Neukirch J., Schmidt A., Wingberg K.: Cohomology of Number Fields. 2nd edn. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Soulé C.: K-theorie des anneaux d’entiers de corps de nombres et cohomologie étale.. Invent. Math. 55, 251–295 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Tamagawa A.: The Grothendieck conjecture for affine curves. Compos. Math. 109, 135–194 (1997)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Technische Universität MünchenGarching bei MünchenGermany

Personalised recommendations