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On the Section Conjecture for Torsors

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

Abstract

We explore the weak section conjecture for torsors W under a connected algebraic group Gk: we ask whether a torsor is necessarily trivial if its extension \({\pi }_{{}_{1}}(W/k)\) splits. Continuous Kummer theory, or a direct construction using neighbourhoods, shows that \({\pi }_{{}_{1}}(W/k)\) splits if and only if the torsor W is compatibly divisible by étale isogenies, see Corollary 175. Therefore the weak section conjecture holds for torsors under tori and, if G is an abelian variety, is linked to Bashmakov’s problem. As an application, we study the relative Brauer group of hyperbolic curves with completion of genus 0 and show a weak section conjecture for such curves, see Proposition 187.Torsors under algebraic groups occur in the context of the section conjecture for a smooth projective curve as the universal Albanese torsor \({\mathrm{Alb}}_{X}^{1} ={ \mathrm{Pic}}_{X}^{1}\) and via the abelianization of sections, see Definition 26. Zero cycles on a smooth projective variety Xk can be considered as an abelianized version of a rational point. We construct a map that assigns to a zero cycle z of degree 1 on X a section s z of the abelianized fundamental group extension \({\pi }_{1}^{\mathrm{ab}}(X/k)\) that lifts the corresponding section of the rational point of the universal Albanese torsor corresponding to the zero cycle.We work over a field k of characteristic 0.

Keywords

Algebraic Group Abelian Variety Smooth Projective Variety Profinite Group Algebraic Number Field 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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