Supersolvable descent for rational points

TL;DR

This paper develops a new descent theory replacing algebraic tori with finite supersolvable groups, and applies it to show density of rational points in certain quotient spaces, leading to new Galois extension results.

Contribution

It introduces a novel descent framework for supersolvable groups and applies it to rational points and Galois extensions, extending previous theories.

Findings

Rational points are dense in the Brauer-Manin set for certain quotients.

Existence of supersolvable Galois extensions with prescribed norms.

Generalization of previous work by Frei-Loughran-Newton.

Abstract

We construct an analogue of the classical descent theory of Colliot-Th\'el\`ene and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer-Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei-Loughran-Newton.