Galois Deformation Ring and Barsotti-Tate Representations in the Relative Case

Abstract

In this thesis, we study finite locally free group schemes, Galois deformation rings, and Barsotti-Tate representations in the relative case. We show three independent but related results, assuming p > 2.

First, we give a simpler alternative proof of Breuil’s result on classifying finite flat group schemes over the ring of integers of a p-adic field by certain Breuil modules [5]. Second, we prove that the locus of potentially semi-stable representations of the absolute Galois group of a p-adic field K with a specified Hodge-Tate type and Galois type cuts out a closed subspace of the generic fiber of a given Galois deformation ring, without assuming that K/Qp is finite. This is an extension of the corresponding result of Kisin when K/Qp is finite [19]. Third, we study the locus of Barsotti-Tate representations in the relative case, via analyzing certain extendability of p-divisible groups. We prove that when the ramification index is less than p-1, the locus of relative Barsotti-Tate representations cuts out a closed subspace of the generic fiber of a Galois deformation ring, if the base scheme is 2-dimensional satisfying some conditions. When the ramification index is greater than p-1, we show that such a result does not hold in general.