The Breuil--Mézard conjecture when $l\ne p$

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 $\bar{\rho }:G_F\to G{L}_n(\mathbb{F})$ be a continuous representation. Let $R^{\square }(\bar{\rho })$ be the universal framed deformation ring for $\bar{\rho }$. If $l=p$, then the Breuil--Mézard 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 $G{L}_n({\mathcal{O}}_F)$. We state an analogue of the Breuil--Mézard conjecture when $l\ne p$, 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.

56 pages. To appear in Duke Math. Journal

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