The Breuil--M\'{e}zard conjecture when $l \neq p$

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 $\bar{\rho} : G_F \rightarrow GL_n(\mathbb{F})$ be a continuous representation. Let $R^\square(\bar{\rho})$ be the universal framed deformation ring for $\bar{\rho}$. If $l = p$, then the Breuil--M\'{e}zard conjecture (as formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^\square(\bar{\rho})$ to the mod $l$ reduction of certain representations of $GL_n(\mathcal{O}_F)$. We state an analogue of the Breuil--M\'{e}zard conjecture when $l \neq p$, and prove it whenever $l > 2$ using automorphy lifting theorems. We give a local proof when $l$ is "quasi-banal" for $F$ and $\bar{\rho}$ is tamely ramified. We also analyse the reduction modulo $l$ of the types $\sigma(\tau)$ defined by Schneider and Zink.