Number Theory Seminar

The Breuil-Mezard conjecture predicts the geometry of special fibers of local Galois deformations with p-adic Hodge theory condition in terms of modular representation theory. When K = Qp, the conjecture predicts the existence of a special cycles on the unrestricted local deformation ring associated to irreducible representations of GL_n(F_p). I will describe joint work in progress with Daniel Le, Bao V. Le Hung, and Stefano Morra where we construct these cycles in generic situations for arbitrary n and prove the conjecture for a certain class of potentially crystalline deformation rings.