Abstract
We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called ‘Taylor–Wiles hypothesis’. We apply this to the problem of the modularity of elliptic curves over totally real fields.
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Acknowledgments
During the period this research was conducted, Jack Thorne served as a Clay Research Fellow. I would like to thank Chandrashekhar Khare for inspiring conversations, and for showing me his papers [26, 27]. I would also like to thank Toby Gee and David Geraghty for useful conversations, and the anonymous referee for their detailed comments and corrections.
