Kisin Varieties Associated to Reducible Galois Representations of Dimension 2

Abstract

We consider a semi-module decomposition of the Kisin variety associated to a 2-dimensional mod p reducible Galois representation ρ of a p-adic field K and a cocharacter μ. Upon associating a tuple of matrices b to the representation ρ, there is a group theoretic description of the geometric points of the Kisin variety associated to ρ and μ, which we use to study these semi-modules. In order to do this, we note that b is determined up to a conjugation action and choose a representative b which allows us to compute the semi-modules and identify a finite set of cocharacters which correspond to the potentially nonempty semi-modules. We will also see that, for good choices of b and μ, all nonempty semi-modules are isomorphic to affine spaces.