Hilbert-Samuel multiplicities of certain deformation rings

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.

12 pages; Prop. 5 and Lemma 2 are new, showing that the special fibre of the universal deformation ring is geometrically irreducible and generically reduced