Abstract
We explore the weak section conjecture for torsors W under a connected algebraic group G ∕ k: 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 X ∕ k 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
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Stix, J. (2013). On the Section Conjecture for Torsors. 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_13
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DOI: https://doi.org/10.1007/978-3-642-30674-7_13
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