Wnasu Kim (Cambridge)

Galois deformation theory for norm fields

In this talk, we give a variant of the proof of Kisin's connected component analysis of a certain flat deformation rings, which is one of the key technical steps in his proof of modularity lifting theorem for potentially Barsotti-Tate representations. The variant that will be presented at the talk does not use the ``Breuil-Kisin classification of finite flat group schemes,'' and it does not take a serious separate input to treat the case $p=2$. Note that to prove the classification of connected finite flat group schemes over $2$-adic dvr one needs Zink's theory of windows and displays, and Khare-Wintenberger's proof of Serre's conjecture uses the case $p=2$ of potentially Barsotti-Tate modularity lifting theorem.
The new ingredient in this variant is the Gal(Kbar/Kinfty)-deformation theory. Let K be a finite extension of $Qp, and choose a uniformizer pi of K. Choose $\pi_{n+1}:=\sqrt[p^n]{\pi}$ such that $\pi_{n+1}^p=\pi_n$, and put $K_\infty:=\bigcup_n K(\pi_{n+1})$. In this talk, we introduce a new deformation ring for $Gal(\overline K/K_\infty)$-representations of ``height $\leqslant h$'' for some positive integer $h$, which naturally maps into crystalline and semi-stable deformation rings with Hodge-Tate weights in $[0,h]$ (in the sense of Kisin). We will also discuss how to use this $Gal(\overline K/K_\infty)$-deformation ring of ``height $\leqslant 1$'' to deduce Kisin's result on flat deformation rings (hence his modularity lifting theorem for potentially Barsotti-Tate representations).