Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$

Abstract

We study the rigid generic fiber ${\mathcal{X}}_{\overline{\rho }}^{\square }$ of the framed deformation space of the trivial representation $\overline{\rho }:G_K\to {\text{GL}}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/{\mathbf{Q}}_p$. Under some mild conditions on $K$ we prove that ${\mathcal{X}}_{\overline{\rho }}^{\square }$ is normal. When $p>n$ we describe its irreducible components, and show Zariski density of its crystalline points.

Updated numbering to match published version