The Geometry of Unipotent Deformations and Applications

This thesis studies primarily the local properties the unipotent connected component of the moduli space of Langlands parameters, the local rings of which give us Galois deformation rings, a crucial ingredient in the Taylor-Wiles-Kisin patching method that is used to prove global Langlands correspondences. We study first the simpler ‘considerate’ case to give a criterion for smoothness of the connected components when G = GLn. We also study the local rings of various unions of connected components to show that the Galois deformation rings are Cohen-Macaulay. We study further the Steinberg component in the case of ‘extreme inconsiderateness’ to show that the Steinberg component has at most rational singularities, so in particular is normal and Cohen-Macaulay. Finally, we give an application of the smoothness result, to give a freeness result of the module of certain Hida families of automorphic forms over its Hecke algebra, which in turn will give a multiplicity result for the Galois representations of these Hida families.