Congruences between Hilbert modular forms: constructing ordinary lifts

By Thomas Barnet-Lamb, T. Gee, David Geraghty. 2010

54 citations recorded by the source corpus.

Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each place dividing l. We deduce a similar result for r itself, under the assumption that at places v|l the representation r|_{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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3ae346732829db6a25f31b9b3f15971276a1c31b

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In Geometry and Topology

1. T. Gee
2. David Savitt
3. M. Emerton
4. David Geraghty
5. Thomas Barnet-Lamb
6. Tongyin Liu
7. Bao V. Le Hung
8. Nuno Freitas

In Mathematical Physics

1. T. Gee
2. David Savitt
3. M. Emerton
4. David Geraghty
5. Thomas Barnet-Lamb
6. Tongyin Liu
7. M. Kisin
8. J. Thorne

Citations in the corpus

• Elliptic curves over real quadratic fields are modular (2013, background): “[3] T. Barnet-Lamb, T. Gee and D. Geraghty, Congruences between Hilbert modular forms: constructing ordinary lifts, Duke Math.”
• The buzzard-diamond-jarvis conjecture for unitary groups (2012, background): “The following lemma is a slight variant of [BLGG12, Lem.”
• Serre weights for rank two unitary groups (2011, methodology): “1 of [BLGG10], which proves an analogous result over totally real fields.”
• A geometric perspective on the Breuil–Mézard conjecture (2011, background): “(for example, any potentially Barsotti–Tate lift is potentially diagonalizable, and the result is then immediate in the irreducible case, and in the reducible case is a special case of Lemma 6.1.6 of [BLGG12]). We anticipate that a similar argument works more generally. A.5. Lemma. Let p∤ 2nbe prime”
• THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS (2012, unknown): “ology of [Sno09].) If ρ|G L + v is irreducible, this is proved in the course of the proof of Proposition 7.8.1 of [Sno09], and if it is reducible then it is an immediate consequence of Lemma 6.1.6 of [BLGG12]. Combining Proposition A.1 and Theorem A.2, we obtain the following result. Corollary A.3. ”