The Breuil–Mézard conjecture when l≠p

Abstract

Let $l$ and $p$ be primes, let $F/{\mathbb{Q}}_p$ be a finite extension with absolute Galois group $G_F$, let $\mathbb{F}$ be a finite field of characteristic $l$, and let

$$\overline{\rho }:G_F\to {GL}_n\left(\mathbb{F}\right)$$ be a continuous representation. Let $R^{\square }\left(\overline{\rho }\right)$ be the universal framed deformation ring for $\overline{\rho }$. If $l=p$, then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^{\square }\left(\overline{\rho }\right)$ to the mod $l$ reduction of certain representations of ${GL}_n\left({\mathcal{O}}_F\right)$. We state an analogue of the Breuil–Mézard conjecture when $l\ne p$, and we prove it whenever $l>2$ using automorphy lifting theorems. We give a local proof when $l$ is “quasibanal” for $F$ and $\overline{\rho }$ is tamely ramified. We also analyze the reduction modulo $l$ of the types $\sigma \left(\tau \right)$ defined by Schneider and Zink.