Galois deformation theory for norm fields and flat deformation rings

Abstract

Let K be a finite extension of Q(p), and choose a uniformizer pi is an element of K, and put K(infinity) := K((p infinity)root pi). We introduce a new technique using restriction to Gal((K) over bar /K(infinity)) to study flat deformation rings. We show the existence of deformation rings for Gal((K) over bar /K(infinity))-representations "of height <= h" for any positive integer h, and prove that when h = 1 they are isomorphic to "flat deformation rings". This Gal((K) over bar /K(infinity))-deformation theory has a good 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. (C) 2011 Elsevier Inc. All rights reserved.