Correspondance de Langlands locale $p$-adique et anneaux de Kisin

We use a $\mathcal{B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$ to give a construction of Kisin’s rings and the attached universal Galois representations (in dimension 2 and for ${\mathbf{Q}}_p$) directly from the classical Langlands correspondence. This gives, in particular, a uniform proof of the geometric Breuil-M'ezard conjecture in the supercuspidal case.