Murphy's Law for Galois Deformation Rings

Abstract

In this paper, we prove, under a technical assumption, that any semi-direct product of a $p$-group $G$ with a group $\Phi$ of order prime to $p$ can appear as the Galois group of a tower of extensions $H/K/F$ with the property that $H$ is the maximal pro-$p$ extension of $K$ that is unramified everywhere, and $\operatorname{Gal}(H/K) = G$. A consequence of this result is that any local ring admitting a surjection to $\mathbb{Z}_5$ or $\mathbb{Z}_7$ with finite kernel can occur as a universal everywhere unramified deformation ring.