Irreducibility of some crystalline loci with irregular Hodge--Tate weights

Abstract

We show that loci of crystalline representations of $G_K$ for $K/\mathbb{Q}_p$ an unramified extension are irreducible when the Hodge--Tate weights are fixed and sufficiently small. This was previously known for weights in the interval $[-p,0]$ and in this paper we show how that this bound can be relaxed provided the Hodge--Tate weights are sufficiently irregular at certain embeddings. This is motivated by the desire to extend the conjectures of Breuil--M\'ezard on loci of potentially crystalline representations to irregular weights.