Manuscripta Mathematica

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

On some anabelian properties of arithmetic curves

  • A. IvanovEmail author


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 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Technische Universität MünchenGarching bei MünchenGermany

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