Galois representations and the tame inverse Galois problem

Abstract

In this paper we will focus on a variant of the Inverse Galois Problem over the rationals, emphasizing the progress made through the analysis of the Galois representations arising from arithmetic-geometric objects. The study of the Inverse Galois Problem explores the finite quotients of the absolute Galois groupGQ = Gal(Q=Q), and sheds light on its structure. The Inverse Galois Problem over Q, first considered by D. Hilbert around 1890, asks whether, given a finite groupG, there exists a finite Galois extensionK=Q with Galois groupG. This problem, which remains unsolved today, has given rise to significant mathematical advances, and several different techniques have been developed to address it. A strategy to deal with it is the constructive Galois theory (or rigidity method), which roughly consists in realizing the group G as the Galois group of a polynomial with coefficients in the complex numbers (which is possible thanks to the Riemann existence theorem for compact surfaces) and imposing some conditions that guarantee that, in fact, the polynomial is defined over Q(T), where T is a variable. This method has been very successful, and thanks to it many simple groups are now known to be realizable as Galois groups over Q. For instance, all sporadic simple groups save the Mathieu group M23 have been realized through this method (cf.