Automorphy lifting with adequate image

Automorphy lifting theorems establish situations in which Galois representations over \bar{Q_p} are automorphic if their residual representation has an automorphic lift. In 2018, Allen et. al. proved the first automorphy lifting theorem for n-dimensional Galois representations over a CM field without any self-duality assumptions. Their theorem imposes a condition on the image of the residual representation, requiring it to be ``enormous’’. In this talk, we discuss how to relax this big image assumption. Based on joint work with Jack Thorne.