Duke Mathematical Journal

Congruences between Hilbert modular forms: constructing ordinary lifts

Thomas Barnet-Lamb, Toby Gee, and David Geraghty

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Abstract

Under mild hypotheses, we prove that if $F$ is a totally real field, and $\overline{\rho}:G_{F}\to\operatorname{GL}_{2}(\overline{\mathbb {F}}_{l})$ is irreducible and modular, then there is a finite solvable totally real extension $F^{\prime}/F$ such that $\overline{\rho}|_{G_{F^{\prime}}}$ 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.

Article information

Source
Duke Math. J. Volume 161, Number 8 (2012), 1521-1580.

Dates
First available in Project Euclid: 22 May 2012

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1337690407

Digital Object Identifier
doi:10.1215/00127094-1593326

Zentralblatt MATH identifier
06050950

Mathematical Reviews number (MathSciNet)
MR2931274

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]

Citation

Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David. Congruences between Hilbert modular forms: constructing ordinary lifts. Duke Math. J. 161 (2012), no. 8, 1521--1580. doi:10.1215/00127094-1593326. http://projecteuclid.org/euclid.dmj/1337690407.


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