The Breuil-Mézard conjecture when l is not equal to p

Abstract

Let l and p be primes, let F/Q_p be a finite extension with absolute Galois group G_F, let F be a finite field of characteristic l, and let p̄ : G_F→ GL_n(F) be a continuous representation. Let R^□(p̄) be the universal framed deformation ring for p̄. If l = p, then the Breuil-Mézard conjecture relates the mod l reduction of certain cycles in R^□(p̄) to the mod l reduction of certain representations of GL_n(O_F). We give an analogue of the Breuil-Mézard conjecture when l ≠ p, and prove it whenever l > 2 using automorphy lifting theorems. We also give a local proof when n = 2 and l> 2 by explicit calculation, and also when l is "quasi-banal'' for F and p̄ is tamely ramified.