Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Abstract

Abstract We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation ρ ¯ {\bar{\rho}} of a p-adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds if K is totally ramified with n = 3 {n=3} or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ ¯ {\bar{\rho}} of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.