Galois deformation theory for norm fields and flat deformation rings
Abstract
Let K be a finite extension of Qp, and choose a uniformizer π∈ K, and put K∞:=K([p∞]π). We introduce a new technique using restriction to ( K/K∞) to study flat deformation rings. We show the existence of deformation rings for ( K/K∞)-representations ``of height ≤slant h'' for any positive integer h, and we use them to give a variant of Kisin's proof of connected component analysis of a certain flat deformation rings, which was used to prove Kisin's modularity lifting theorem for potentially Barsotti-Tate representations. Our proof does not use the classification of finite flat group schemes, so it avoids Zink's theory of windows and displays when p=2. This ( K/K∞)-deformation theory has a good analogue in positive characteristics analogue of crystalline representations in the sense of Genestier-Lafforgue. In particular, we obtain a positive characteristic analogue of crystalline deformation rings, and can analyze their local structure.