Deformations of p-adic Galois representations

We present the numerical and geometric conjecture of modular multiplicities due to Breuil-M., Kisin and Emerton-Gee, and its influence on the p-adic Langlands program. The p-adic Hodge theory gives an explicit way to approach this conjecture and reveals new arithmetic phenomena. We derive from this explicit presentations of some potentially Barsotti–Tate deformation rings.