Solvable Descent and the Grunwald Problem for Solvable Groups

TL;DR

This paper proves a fibration theorem over quasi-trivial tori that implies solvable descent, providing a positive solution to the Grunwald problem for solvable groups and an alternative proof of Shafarevich's result.

Contribution

It introduces a new fibration approach that generalizes Shafarevich's positive solution to the Inverse Galois Problem for solvable groups, avoiding his traditional shrinking method.

Findings

Positive answer to the Grunwald problem for solvable groups up to Brauer--Manin obstruction

Provides an alternative proof of Shafarevich's result on the Inverse Galois Problem

Computes the Brauer--Manin obstruction as a linear combination of Redéi symbols

Abstract

We prove a suitable fibration theorem over quasi-trivial tori that, through an approach developed by Harpaz and Wittenberg, implies so-called solvable descent. In particular, this gives a positive answer to the Grunwald problem for solvable groups up to the necessary Brauer--Manin obstruction, providing a generalizion of Shafarevich's positive answer to the Inverse Galois Problem for solvable groups. This also provides an alternative proof of Shafarevich's result that avoids his "shrinking procedure". For the fibration theorem, we first adapt the starting ideas of Shafarevich for the creation of local lifts. To deal then with the Brauer--Manin obstruction (i.e. the relevant local-to-global obstruction), we compute its "triple variation" on grids of fibers. The resulting expression is a linear combination of Red\'ei symbols on the base. Customizing these and employing a combinatorial principle first noted by Alexander Smith in the context of Class and Selmer Groups, one infers the vanishing of the obstruction in at least one fiber.