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

TL;DR

This paper constructs Kisin's rings and universal Galois representations for ${ m GL}_2({f Q}_p)$ using the $p$-adic local Langlands correspondence, providing a uniform proof of the Breuil-Mézard conjecture in the supercuspidal case.

Contribution

It offers a novel construction of Kisin's rings directly from the classical Langlands correspondence, linking local and $p$-adic Langlands theories.

Findings

Constructed Kisin's rings from classical Langlands correspondence.

Provided a uniform proof of the geometric Breuil-Mézard conjecture.

Connected $p$-adic and classical Langlands frameworks.

Abstract

We use a ${\mathcal B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm {GL}}_2({\mathbf Q}_p )$ 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.