A Modularity Lifting Theorem for Weight Two Hilbert Modular Forms

By T. Gee. 2006

49 citations recorded by the source corpus.

We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/61550f5162ab8ef89d0453597c8415f867bb6a62

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In Geometry and Topology

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

In Mathematical Physics

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

Citations in the corpus

• On the weights of mod p Hilbert modular forms (2006, methodology): “In fact, Kisin’s theorem is not general enough for our applications, and we need to use the main theorem of [Gee06].”
• THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS (2012, methodology): “The main idea of the present paper is that one can use the modularity lifting theorems proved in [Kis09b] and [Gee06] (by a completely different method, unrelated to p-adic local Langlands) to prove the Breuil–Mézard conjecture for all potentially Barsotti–Tate deformation rings.”
• Congruences between Hilbert modular forms: constructing ordinary lifts (2010, methodology): “To see that we can do this, note that by Propositions 6.1.3 and 6.1.4, there is a finite solvable extension of totally real fields F+1 /F + and a regular algebraic ι-ordinary cuspidal automorphic representation π1 of GL2(AF+1 ) of weight 0 such that • π1 is unramified at all finite places.”
• General Serre weight conjectures (2015, unknown): “By the results of [Kis09c, Gee06] potential diagonalisability is known for the system BT, and Proposition 4.2.1 (applied to the Taylor–Wiles–Kisin patching method for automorphic forms on suitable quaternion algebras or forms of U(2)) then implies the result of [GK14] discussed in Example 3.3.9.”
• Automorphic lifts of prescribed types (2008, methodology): “To our knowledge, the only such theorems in the literature are those for weight two Hilbert modular forms in [Kis07c] and [Gee06a], and the work of [Kis06] for modular forms over totally real fields in which p splits completely.”
• Lattices in the cohomology of Shimura curves (2013, methodology): “…6) could be proved without relying on the main results of [GK12], and without using the potential diagonalizability of potentially Barsotti–Tate representations proved in [Kis09b] and [Gee06], but rather just the explicit computations of potentially Barsotti–Tate deformation rings in Section 7.”
• Serre weights for rank two unitary groups (2011, methodology): “From the existence of ρ1 and ρ2, Proposition 2.3 of [Gee06], and Corollary 2.5.16 of [Kis07], it follows that • For each place v|l, rl,ı(π ′′ 1 )|GF1,v is diagonalizable.”
• The Sato-Tate conjecture for Hilbert modular forms (2009, unknown): “Then rχ′ |G F + w ∼ rl,ι(π′)|G F + w , because both representations are Barsotti-Tate and non-ordinary (see Proposition 2.3 of [Gee06], which shows that all non-ordinary points lie on the same component of the appropriate lifting ring).”