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Congruences between Hilbert modular forms: constructing ordinary lifts. (English) Zbl 1297.11028
Summary: Under mild hypotheses, we prove that if $$F$$ is a totally real field, and $$\overline{\rho}:G_{F}\to\mathrm{GL}_2(\overline{\mathbb F}_l)$$ is irreducible and modular, then there is a finite solvable totally real extension $$F'/F$$ such that $$\overline{\rho}|_{G_{F'}}$$ 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 $$\overline {\rho}|_{G_{F_v}}$$ 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.

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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