Hilbert-Samuel multiplicities of certain deformation rings

We compute presentations of crystalline framed deformation rings of a twodimensional representation $\overline{\rho}$ of the absolute Galois group of $\mathbb{Q}_p$, when $\overline{\rho}$ has scalar semi-simplification, the Hodge-Tate weights are small and $p \gt 2$. 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 $\overline{\rho}$. We show that in the last two cases the deformation ring is not Cohen-Macaulay.

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Published 13 October 2014