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

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.

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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