Solvable Descent and the Grunwald Problem for Solvable Groups

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.