Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

TL;DR

This paper investigates the connectedness of Kisin varieties linked to irreducible Galois representations, confirming Kisin's conjecture in specific cases and providing counterexamples in general.

Contribution

It proves Kisin's conjecture for certain cases and demonstrates that it does not hold universally, also deriving related connectedness results for deformation rings.

Findings

Kisin's conjecture holds when $K$ is totally ramified with $n=3$

Counterexamples show the conjecture does not hold in general

Connectedness results for deformation rings are established

Abstract

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