Galois deformation theory for norm fields and flat deformation rings

Let K be a finite extension of Qp, and choose a uniformizer π∈K, and put . We introduce a new technique using restriction to to study flat deformation rings. We show the existence of deformation rings for -representations “of height ≤h” for any positive integer h, and prove that when h=1 they are isomorphic to “flat deformation rings”. This -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.