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.