Seminar on Breuil-Mezard conjecture

organizers: Przemyslaw Chojecki, Gabriel Dospinescu and Benjamin Schraen 

Date: March - April 2013, each Wednesday at 14.00h (exceptions listed below); there are two 90-minutes talks.
Place: 4, place Jussieu (Paris, France); room 417, corridor 15-16  (4th floor)

Description of the seminar:
    The Breuil-Mezard conjecture, which appears for the first time in [BM1], predicts the Hilbert-Samuel multiplicity of the special fibre of a deformation ring of a mod p local Galois representation. Originally it concerns only GL_2(Q_p), but the conjecture is now believed to be true for other groups as well. Recent developments show that the conjecture lies at the intersection of the p-adic Local Langlands conjecture, modularity lifting theorems and completed cohomology of Shimura varieties.
    The goal of the seminar is to understand two different proofs of the conjecture. The first one, by Paskunas [Pa], uses heavily known facts about the p-adic Local Langlands correpondence for GL_2(Q_p). The second one, by Gee-Kisin [GK], uses modularity lifting techniques and gives a proof for GL_2(F), where F is a finite extension of Q_p. Also, we will want to understand how the geometric perspective on the conjecture (see [EG] and [BM2]) can be applied to get the results about the completed cohomology of Shimura curves, which is done in [EGS].       


[BM1] C. Breuil, A. Mezard, "
Multiplicités modulaires et représentations de GL_2(Z_p) et de Gal(\bar{Q_p}/Q_p) en l=p"
[BM2] C. Breuil, A. Mezard, "Multiplicités modulaires raffinées"
[EG] M. Emerton, T. Gee, "A geometric perspective on the Breuil-Mézard conjecture"
[EGS] M. Emerton, T. Gee, D. Savitt, "Lattices in the cohomology of Shimura curves", work in preparation
[GK] T. Gee, M. Kisin, "
The Breuil-Mézard conjecture for potentially Barsotti-Tate representations"
[Pa] V. Paskunas, "On the Breuil-Mézard conjecture"



Talk 1 (Benjamin Schraen): Introduction to the Breuil-Mezard conjecture. Applications. Geometric version of the conjecture after [BM2] and [EG].

Talk 2 (Przemyslaw Chojecki): Proof of geometric version in dimension 2 and n-dimensional Breuil-Mezard conjecture (sections 3.2-3.4 and chapter 4, [EG]). 


Talk 1 (Benjamin Schraen): Reminder on p-adic local Langlands correspondence. Resume of results of Paskunas on projective envelopes (P^\tilde). Results of Colmez on algebraic vectors.

Talk 2 (Eugen Hellmann): Study of M(\Theta) module of Paskunas. Proof of proposition 2.20 of [Pa].


Talk 1 (Gabriel Dospinescu): Projectivity of module P^\tilde in the generic case (chapter 5, [Pa]) et consequences for reduction of M(\Theta).

Talk 2 (Gabriel Dospinescu): Algebraic vectors and end of the proof of [Pa].


Talk 1 (Stephane Bijakowski): Deformation rings and algebraic automorphic forms (chapters 2 and 3, [GK]).

Talk 2 (Przemyslaw Chojecki): Kisin-Taylor-Wiles patching (sections 4.1-4.4, [GK]).


Talk (Benjamin Schraen): Continuation on patching and lifting of local representations (section 4.5 and Appendix, [GK]).


Talk (Przemyslaw Chojecki): On patching functors and generic BDJ conjecture (after [EGS]).

The last meeting of the seminar.