On the Section Conjecture for Torsors

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.