Congruences between Hilbert modular forms: constructing ordinary lifts

Abstract

Under mild hypotheses, we prove that if $F$ is a totally real field, and $\overline{\rho }:G_F\to {GL}_2\left({\overline{\mathbb{F}}}_l\right)$ is irreducible and modular, then there is a finite solvable totally real extension $F\prime /F$ such that $\overline{\rho }{\mid }_{G_{F^{\prime }}}$ has a modular lift which is ordinary at each place dividing $l$. We deduce a similar result for $\overline{\rho }$ itself, under the assumption that at places $v|l$ the representation $\overline{\rho }{\mid }_{G_{F_v}}$ is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti–Tate representations and the Buzzard–Diamond–Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank $4$ unitary groups.