On the potential automorphy of certain odd-dimensional Galois representations

Abstract In a previous paper, the potential automorphy of certain Galois representations to GL n for n even was established, following the work of Harris, Shepherd–Barron and Taylor and using the lifting theorems of Clozel, Harris and Taylor. In this paper, we extend those results to n =3 and n =5, and conditionally to all other odd n . The key additional tools necessary are results which give the automorphy or potential automorphy of symmetric powers of elliptic curves, most notably those of Gelbert, Jacquet, Kim, Shahidi and Harris.