# Galois deformation theory for norm fields and flat deformation rings - Mathematics > Number Theory

Galois deformation theory for norm fields and flat deformation rings - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let $K$ be a finite extension of $\mathbb{Q} p$, and choose a uniformizer$\pi\in K$, and put $K \infty:=K\sqrtp^\infty{\pi}$. We introduce a newtechnique using restriction to $\Gal\ol K-K \infty$ to study flat deformationrings. We show the existence of deformation rings for $\Gal\olK-K \infty$-representations ``of height $\leqslant h$- for any positiveinteger $h$, and we use them to give a variant of Kisin-s proof of connectedcomponent analysis of a certain flat deformation rings, which was used to proveKisin-s modularity lifting theorem for potentially Barsotti-Taterepresentations. Our proof does not use the classification of finite flat groupschemes, so it avoids Zink-s theory of windows and displays when $p=2$.This $\Gal\ol K-K \infty$-deformation theory has a good analogue inpositive characteristics analogue of crystalline representations in the senseof Genestier-Lafforgue. In particular, we obtain a positive characteristicanalogue of crystalline deformation rings, and can analyze their localstructure.

Author: ** Wansu Kim**

Source: https://arxiv.org/