Zariski density of crystalline points on $\GSp_{2n}$-valued local deformation rings

TL;DR

This paper introduces trianguline deformation spaces for GSp(2n)-valued Galois representations and proves that crystalline points are Zariski dense in these spaces under certain conditions.

Contribution

It establishes the Zariski density of crystalline points on deformation spaces of GSp(2n)-valued Galois representations, extending understanding of their geometric structure.

Findings

Crystalline points are Zariski dense in the deformation space.

Properties of trianguline deformation spaces are analyzed.

Conditions for density are identified.

Abstract

We introduce trianguline deformation spaces $X_{\tri} (\overline{\rho})$ for a $\GSp_{2n}$-valued residual representation $\overline{\rho} \colon {\mathcal{G}}_K \to \GSp_{2n} (k)$, where $k$ is a finite field of characteristic $p > 0$ and ${\mathcal{G}}_K$ is the absolute Galois group of a finite extension $K / {\mathbb{Q}}_p$, and study their properties. We show Zariski density of the crystalline points on the rigid generic fiber ${\mathfrak{X}}_{\overline{\rho}}$ of the framed deformation space of $\overline{\rho}$ under some conditions.