Under mild hypotheses, we prove that if F is a totally real field, k is the
algebraic closure of the finite field with l elements and r : G_F --> GL_2(k)
is irreducible and modular, then there is a finite solvable totally real
extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each
place dividing l. We deduce a similar result for r itself, under the assumption
that at places v|l the representation r|_{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.Comment: 48 page
