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

TL;DR

This paper establishes conditions under which two-dimensional mod p automorphic Galois representations over totally real fields admit automorphic crystalline lifts, highlighting the role of local restrictions and obstructions related to dihedral representations.

Contribution

It provides a sufficient condition for the existence of automorphic crystalline lifts of mod p Galois representations, emphasizing the importance of local determinant restrictions and identifying obstructions for certain dihedral cases.

Findings

A sufficient condition for automorphic crystalline lifts involving local determinant restrictions.

Obstructions to controlling level and character are linked to badly dihedral representations.

The main obstruction to lifting is characterized explicitly for certain cases.

Abstract

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