Automorphy for Some \(l\)-Adic Lifts of Automorphic Mod \(l\) Galois Representations

We extend the methods of Wiles and of Taylor and Wiles from \(GL_2\) to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, \(l\)-adic lifts of certain automorphic mod \(l\) Galois representations of any dimension. We also make a conjecture about the structure of mod \(l\) automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for \(GL_2\) . Following Wiles’ method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.Mathematic