Seminar des SFB/TRR 326 GAUS Automorphy Lifting with ˆG-Adequate Image

Let F be a number field and G a (split) reductive group. The Langlands program attempts to relate

1. automorphic representations of G
2. representations of the absolute Galois group of F valued in ˆG, the dual
   group of G.

Automorphy lifting theorems are a way to go from (2) to (1). Such theorems are proved using the Taylor–Wiles method but require certain 'big image' hypotheses.

We discuss a generalization of the Taylor–Wiles method. Our generalization results in a weakened big image condition (ˆG-adequate image), which turns out to hold under certain irreducibility hypotheses. We conclude with applications to modularity of some elliptic curves over quadratic extensions of totally real fields, building on work of Boxer–Calegari–Gee–Pilloni.