Hilbert-Samuel multiplicities of certain deformation rings

TL;DR

This paper computes explicit presentations of certain crystalline deformation rings of 2-dimensional Galois representations, analyzing their geometric properties and Hilbert-Samuel multiplicities under specific conditions.

Contribution

It provides explicit descriptions of crystalline framed deformation rings for scalar semi-simplified representations with small Hodge-Tate weights, including their geometric and multiplicity properties.

Findings

Special fibre is geometrically irreducible and generically reduced.

Hilbert-Samuel multiplicity is 1, 2, or 4 depending on the representation.

Deformation rings with multiplicity 2 or 4 are not Cohen-Macaulay.

Abstract

We compute presentations of crystalline framed deformation rings of a two dimensional representation $\bar{\rho}$ of the absolute Galois group of $\mathbb{Q}_p$, when $\bar{\rho}$ has scalar semi-simplification, the Hodge-Tate weights are small and $p>2$. In the non-trivial cases, we show that the special fibre is geometrically irreducible, generically reduced and the Hilbert-Samuel multiplicity is either $1$, $2$ or $4$ depending on $\bar{\rho}$. We show that in the last two cases the deformation ring is not Cohen-Macaulay.