INdAM Program on Serre Conjectures and the p-adic local Langlands program
It is expected that the mod p completed cohomology groups of Shimura curves realize a hypothetical mod p Langlands correspondence. Breuil and Paskunas constructed combinatorial representation theoretic objects called Diamond diagrams (close relatives of Hecke modules) and conjectured that some of them appear in the cohomology of Shimura curves at first principal congruence level. This conjecture was proven in groundbreaking work of Emerton{Gee{Savitt. We will present a refinement of this result. Namely, we determine the diagram structure of the generic part of this cohomology in terms of the local Galois action. We also prove a conjecture of Breuil relating Diamond diagrams and Galois representations using a combinatorial generalization of Colmez's functor. We will emphasize the role played by the Taylor{Wiles method in the above results. Some of the results discussed are part of joint works with A. Dotto, S. Morra, and B. Schraen and some were obtained independently by Y. Hu and H. Wang.