Galois Deformation Theory for Norm Fields

Let K be a finite extension of Q_p, and choose a uniformizer π in K. Choose π_n+1 a pⁿ-th root of π such that (π_n+1)^p=π_n, and put K_inf equal to the union of the K(π_n+1). Let G denote the absolute Galois group of K_inf. In this talk, I will introduce a new deformation ring for G-representations of ``height <= h'' for some positive integer h, which naturally maps into crystalline and semi-stable deformation rings of Hodge-Tate weights in [0,h] (in the sense of Kisin).

As an application of ``G-deformation rings of height <= h'', one can obtain Kisin's connected component analysis of flat deformation rings (of a certain fixed type) for p>2, which is crucially used in his proof of the modularity lifting of potentially Barsotti-Tate representations. This new proof does not use the Breuil-Kisin classification of finite flat group schemes (which is problematic to prove if p=2), and can be modified to work in the case p=2. Kisin handled the p=2 case by extending the Breuil-Kisin classification to connected finite flat group scheme over a 2-adic base which requires a completely separate treatment using Zink's theory, but the new proof avoids this.