Families of p̳-adic Galois representations

Abstract

In this thesis, I first generalize Kisin's theory of finite slope subspaces to arbitrary p-adic fields, and then apply it to the generic fibers of Galois deformation spaces. I study the finite slope deformation rings in details by computing the dimensions of their Zariski cotangent spaces via Galois cohomologies. It turns out that the Galois cohomologies tell us not only the formal smoothness of finite slope deformation rings, but also the behavior of the Sen operator near a generic de Rham representation. Applying these results to the finite slope subspace of two dimensional Galois representations of the absolute Galois group of a p-adic field, we are able to show that a generic (indecomposible) de Rham representation lies in the finite slope subspace. It follows from the construction of the finite slope subspace that the complete local ring of a point in the finite slope subspace is closely related to the finite slope deformation ring at the same point. As a consequence, we manage to show the flatness of the weight map near generic de Rham points, and accumulation and smoothness of generic de Rham points. In particular, we have a precise dimension formula for the finite slope subspace. Taking into account twists by characters, we define the nearly finite slope subspace, which is believed to serve as the local eigenvariety, as is suggested by Colmez's theory of trianguline representation. Following Gouv~a- Mazur and Kisin, we construct an infinite fern in the local Galois deformation space. Moreover, we define the global eigenvariety for GL2 over any number field, and give a lower bound of its dimension.