Crystalline lifts of two-dimensional mod $p$ automorphic Galois representations

Abstract

We show that a sufficient condition for an irreducible automorphic Galois representation $\rho :G_F\to {\mathrm{GL}}_2({\overline{\mathbf{F}}}_p)$ of a totally real field $F$ to have an automorphic crystalline lift is that for each place $v$ of $F$ above $p$ the restriction $\det \rho {|}_{I_v}$ is a fixed power of the 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.

22 pages