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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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