Noether’s and Grunwald’s problems rationality: approximation and obstructions - C.Demarche
Abstract
The Noether problem asks whether the subfield of invariant functions in the function field of a faithful representation of a finite group is a rational extension of the base field. A positive answer to this question leads to a positive answer to the inverse Galois problem over global fields, in a strong form where one can prescribe the local Galois group at a finite number of places (the so-called Grunwald's problem). We will survey variants of the Noether rationality problem and of the Grunwald weak approximation question, explaining certain obstructions to rationality (e.g. unramified cohomology) or to weak approximation (e.g. Brauer-Manin obstruction). We will mention recent progresses on these questions, including for instance the solution to the Grunwald problem for supersolvable groups or counting results on points of bounded height on varieties associated to the inverse Galois problem.