Remarks on Automorphy of Residually Dihedral Representations

Abstract

We prove automorphy lifting results for geometric representations $\rho :G_F\to G{L}_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of ${\mathbb{Q}}_p$ with $p$ an odd prime, such that the residual representation $\bar{\rho }$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F({\zeta }_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $G{L}_2({\mathbb{F}}_7)$.

arXiv comment10 pages