Connectedness of Kisin varieties for GL_2

TL;DR

This paper proves that Kisin varieties for rank 2 simple φ-modules are connected for any cocharacter, clarifying the structure of deformation rings of 2-dimensional Galois representations.

Contribution

It establishes the connectedness of Kisin varieties for rank 2 simple φ-modules across all cocharacters, advancing understanding of deformation rings.

Findings

Kisin varieties for rank 2 are connected for any cocharacter.

Connected components of the deformation ring correspond to fixed Hodge-Tate weight multiplicities.

Provides a geometric insight into the structure of Galois deformation spaces.

Abstract

We show that the Kisin varieties associated to simple $\phi$-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.