On abelian birational sections

TL;DR

This paper studies sections of a Galois group quotient related to algebraic varieties, linking their existence to the elementary obstruction and providing conditions over number fields and for curves.

Contribution

It establishes a correspondence between sections and elementary obstructions, and analyzes their existence criteria over number fields and for curves.

Findings

Sections correspond to infinite divisions of the elementary obstruction.

Existence of sections over number fields depends on the vanishing of the elementary obstruction.

Vanishing of the elementary obstruction is not preserved under scalar extensions.

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-Th\'el\`ene 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.