Modular Forms and the Inverse Galois Problem

A result of Shimura and Deligne attaches a two-dimensional modular
representation of the absolute Galois group of the rational numbers to
any Hecke eigenform. By varying the modular form, one can realise "many"
of the simple groups $\mathrm{PSL}_2(\mathbb{F}_{p^d})$ as Galois groups over the rationals.

In the talk I describe the general strategy, a surprising application of
Maeda's conjecture as well as a generalisation to symplectic Galois
representations. Parts of the results presented were obtained in
collaboration with Luis Dieulefait and Sara Arias-de-Reyna.