On abelian birational sections

Abstract

For a smooth and geometrically irreducible variety X over a field k, the quotient of the absolute Galois group G_{k(X)} by the commutator subgroup of G_{\bar k(X)} projects onto G_k. We investigate the sections of this projection. We show that such sections correspond to "infinite divisions" of the elementary obstruction of Colliot-Thélène and Sansuc. If k is a number field and the Tate-Shafarevich group of the Picard variety of X is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree 1. Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.

12 pages; added Remark 3.6; final version

References in corpus (3)

• On Albanese torsors and the elementary obstruction
• Galois sections for abelianized fundamental groups
• Remarks on cycle classes of sections of the arithmetic fundamental group

Cited by in corpus (1)

• The cuspidalisation of sections of arithmetic fundamental groups II