The Breuil–Mézard conjecture when l≠p

Abstract

Let l and p be primes, let F/Qp be a finite extension with absolute Galois group GF, let F be a finite field of characteristic l, and let ρ¯:GF→GLn(F) be a continuous representation. Let R□(ρ¯) be the universal framed deformation ring for ρ¯. 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□(ρ¯) to the mod l reduction of certain representations of GLn(OF). We state an analogue of the Breuil–Mézard conjecture when l≠p, and we prove it whenever l>2 using automorphy lifting theorems. We give a local proof when l is “quasibanal” for F and ρ¯ is tamely ramified. We also analyze the reduction modulo l of the types σ(τ) defined by Schneider and Zink.