Resolving moduli spaces of crystalline representations and modularity

In 2004, Kisin proved modularity lifting theorems for two-dimensional Barsotti-Tate representations of totally real fields. A key ingredient in his proof was the construction of resolutions of moduli spaces of crystalline representations of finite extensions of $\mathbb{Q}_p$ using p-adic Hodge-theoretic data.

 

In this talk I will discuss recent joint work with Bao Le Hung and Brandon Levin which extends these results to three-dimensional Galois representations of minimal regular weight. I will begin by recalling some of Kisin's main ideas, before focusing on the role played in our work by certain affine Springer loci inside the affine Grassmannian. In particular, I will indicate how sufficient control of the singularities of these loci, which we obtain for the quasi-minuscule coweight (2,1,0), largely reduces the problem to a dimension estimate.