Descent obstruction and fundamental exact sequence

TL;DR

This paper explores the relationship between descent obstructions, the fundamental exact sequence, and rational points on varieties over number fields, with applications to the Brauer-Manin obstruction and Grothendieck's section conjecture.

Contribution

It introduces a local-global interpolation property for the fundamental group sequence and applies it to various obstructions and conjectures in arithmetic geometry.

Findings

Finite descent obstruction relates to the fundamental group sequence.

Examples of curves where the birational section conjecture holds non-trivially.

Connections established between descent obstructions and the Brauer-Manin obstruction.

Abstract

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of families of curves over number fields, such that the birational section conjecture is true in a non-trivial way.