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

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.