# Connectedness of Kisin varieties for GL_2

Eugen Hellmann

Number | 38 |
---|---|

Author | Eugen Hellmann |

Project | B08 |

Year | 2010 |

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.