Under mild hypotheses, we prove that if is a totally real field, and is irreducible and modular, then there is a finite solvable totally real extension such that has a modular lift which is ordinary at each place dividing . We deduce a similar result for itself, under the assumption that at places the representation 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 unitary groups.
Thomas Barnet-Lamb. Toby Gee. David Geraghty. "Congruences between Hilbert modular forms: constructing ordinary lifts." Duke Math. J. 161 (8) 1521 - 1580, 1 June 2012. https://doi.org/10.1215/00127094-1593326