Connectedness of Kisin varieties for GL_2

Abstract

We show that the Kisin varieties associated to simple $\varphi$-modules of rank $2$ are connected in the case of an arbitrary cocharacter. This proves that the connected components of the generic fiber of the flat deformation ring of an irreducible $2$-dimensional Galois representation of a local field are precisely the components where the multiplicities of the Hodge-Tate weights are fixed.

18 pages, 2 figures