Inverse Galois Theory

The Inverse Galois Problem for a given field K is the following question, going back to Hilbert:

Is every finite group a quotient of the absolute Galois group of K?

While this question is trivially wrong for many fields, e.g. algebraically closed fields, and is known for the rational function field of the complex numbers, it is still open in fundamental case K=Q, or Q(t).

The methods to show that a given finite group is a Galois group over Q or Q(t) often rely on deep results in Algebraic and Arithmetic geometry, like Galois representations associated to motives or automorphic forms or the detection of rational points on moduli spaces of covers, called Hurwitz spaces.

Galois representations associated to motives and automorphic forms

The theory of motives was introduced by Alexander Grothendieck in order to linearize the category of algebraic varieties. A motive in its simplest form can be thought of to be a part of the cohomology of an algebriac variety cut out by projections built from algebraic correspondences. These projectors cut out pieces of etale cohomology groups, which are G_K modules in a natural way.

Using various methods from algebraic geometry, notably convolution methods introduced by Nick Katz, one can contruct families of motives leading to many new realizations of finite group as Galois groups over Q(t) and hence over Q.

Arithmetic Geometry of moduli spaces of curves

The study of the representations of the absolute Galois group of rationals in the étale fundamental group of moduli spaces of curves involves geometric and group theoretic aspects, as well as two complementary profinite and proalgebraic sides. The geometric aspect relies on the different properties of the Deligne-Mumford algebraic stack of moduli spaces (DM and inertia stratifications, field of definiton of irreducible components), whereas the group theoretic aspect relies more strongly on computational properties of braids/mapping class groups and topological properties -- the so called Grothendieck-Teichmüller theory.

Through the Malcev-completion of groups, this topic also embraces the study of the category of mixed-Tate motives and their periods also known as multiple zeta values. This study involves both theoretical and computational aspects of relations as lead from the point of view of the geometry of the moduli spaces of curves. This topic has recently seen some developments in relation with the properties of convolution product on perverse sheaves, as well as in relation with the Algebraic Topology field of Operads.