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).