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# Mathematical Research Letters

## Volume 25 (2018)

### Number 4

### Remarks on automorphy of residually dihedral representations

Pages: 1285 – 1304

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n4.a11

#### Author

#### Abstract

We prove automorphy lifting results for geometric representations $\rho : G_F \to GL_2 (\mathcal{O})$, 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 $\overline{\rho}$ 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 $GL_2 (\mathbb{F}_7)$.

Received 26 July 2016

Accepted 9 March 2017

Published 16 November 2018