Automorphic Galois representations in the inverse Galois problem

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms.
The focus will be on ideas and strategies as well as the obstacles that are preventing us from proving much stronger theorems. In this context, the role of coefficient fields will be particularly
highlighted. Many parts are joint work with Sara Arias-de-Reyna, Luis Dieulefait and Sug-Woo Shin.