Park City lecture notes: around the inverse Galois problem
Olivier Wittenberg
TL;DR
The notes survey the inverse Galois problem from a geometric standpoint, tying Hilbert’s irreducibility, Noether’s problem, and versal torsors to the construction of Galois extensions. The regular inverse Galois problem is treated via monodromy, Hurwitz spaces, and the rigidity method, with a focus on rational realizations for many finite groups, especially simple groups. The third part connects Grunwald’s problem to the Brauer–Manin obstruction and descent, arguing that weak approximation on quotient varieties and rationally connected geometry can account for many positive results, while obstruction theories explain many known negatives. Altogether, the notes articulate a coherent framework—via torsors, moduli spaces, and rationality notions—for proving or understanding Galois realizations over number fields and related settings.
Abstract
The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic geometry in the past century, ranging from Noether's problem (letting X denote the quotient of the affine space by a finite group acting linearly, when is X rational?) to the rigidity method (if X is not rational, does it at least contain interesting rational curves?) and to the arithmetic of unirational varieties (if all else fails, does X at least contain interesting rational points?). The goal of the present notes, which formed the basis for three lectures given at the Park City Mathematics Institute in August 2022, is to provide an introduction to these topics.