A Simpler Approach to a Descent Conjecture of Wittenberg

TL;DR

This paper provides an alternative proof for a descent conjecture related to rationally connected torsors, using Cao's descent formula, for specific cases involving connected linear groups.

Contribution

It offers a new proof of Wittenberg's descent conjecture for certain torsors under connected linear groups, expanding the understanding of weak approximation with Brauer-Manin obstruction.

Findings

Proves the conjecture for specific torsors under connected linear groups.

Utilizes Cao's descent formula to establish the result.

Provides an alternative approach to Wittenberg's original proof.

Abstract

A descent conjecture of Wittenberg [Wit24, Conjecture 3.7.4] predicts that if all the twists of a rationally connected torsor over a smooth base satisfy weak approximation with Brauer-Manin obstruction, then so does the base. We give an alternative proof of Wittenberg's conjecture for certain torsors under connected linear groups via Cao's descent formula.