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# Mathematical Research Letters

## Volume 25 (2018)

### Number 1

### Crystalline lifts of two-dimensional $\mathrm{mod} \: p$ automorphic Galois representations

Pages: 43 – 73

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n1.a3

#### Authors

#### Abstract

We show that a sufficient condition for an irreducible automorphic Galois representation $\rho : G_F \to GL_2(\overline{\mathbf{F}}_p)$ of a totally real field $F$ to have an automorphic crystalline lift is that for each place $v$ of $F$ above $p$ the restriction ${\mathrm{det} \rho \vert}_{I_v}$ is a fixed power of the $\mathrm{mod} \: p$ cyclotomic character. Moreover, we show that the only obstruction to controlling the level and character of such automorphic lifts arises for badly dihedral representations.

#### 2010 Mathematics Subject Classification

Primary 11F33. Secondary 11F41, 20C33.

F.D. was partially supported by a Leverhulme Trust Research Project Grant, and partially by EPSRC Grant EP/L025302/1. D.R. was partially supported by an AMS-Simons Research Travel Grant.

Received 3 January 2017

Accepted 29 May 2017

Published 4 June 2018