CONGRUENCES BETWEEN HILBERT MODULAR FORMS: CONSTRUCTING ORDINARY LIFTS

摘要

Under mild hypotheses, we prove that if F is a totally real field, and P : G_F GL+2(F_l) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that pG_F, has a modular lift which is ordinary at each place dividing I. We deduce a similar result for p itself under the assumption that at places v l I the representation plG_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.