Murphy's Law for Galois Deformation Rings

TL;DR

This paper demonstrates that under certain conditions, complex Galois groups can be realized as Galois groups of specific extension towers, and that certain local rings can serve as universal deformation rings in number theory.

Contribution

It proves the realization of semi-direct product Galois groups as maximal unramified pro-p extensions and characterizes local rings as universal deformation rings.

Findings

Any semi-direct product of a p-group with a prime-to-p group can be realized as a Galois group.

Certain local rings can be realized as universal unramified deformation rings.

The results depend on a technical assumption related to Galois extensions.

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.