Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Abstract

AbstractWe consider the Kisin variety associated to ann-dimensional absolutely irreducible modpGalois representationρ¯{\bar{\rho}}of ap-adic fieldKtogether with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds ifKis totally ramified withn=3{n=3}or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated toρ¯{\bar{\rho}}of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.