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.

arXiv commentUpdated numbering to match published version

References in corpus (6)

• Une interprétation modulaire de la variété trianguline
• A local model for the trianguline variety and applications
• The Breuil--Mézard conjecture when $l\ne p$
• Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline lifts
• Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$
• Density of automorphic points in deformation rings of polarized global Galois representations

Cited by in corpus (4)

• On local Galois deformation rings
• Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$
• Zariski density of crystalline points
• Finiteness properties of the category of mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$