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

Abstract

We study the rigid generic fiber $\mathcal{X}^\square_{\overline\rho}$ 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 $\mathcal{X}^\square_{\overline\rho}$ is normal. When $p > n$ we describe its irreducible components, and show Zariski density of its crystalline points.