Zariski density of crystalline points

TL;DR

This paper proves that crystalline points are densely distributed in deformation spaces of Galois representations over p-adic fields, with the result holding locally for all such fields and residual representations.

Contribution

It establishes the Zariski density of crystalline points in deformation spaces, including fixed determinant subspaces, using a purely local approach applicable to all p-adic fields.

Findings

Crystalline points are Zariski dense in deformation spaces.

Density holds in subspaces with fixed crystalline determinant.

Proof is purely local and universally applicable.

Abstract

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a $p$-adic field. We also show that these points are dense in the subspace parameterizing deformations with determinant equal to a fixed crystalline character. Our proof is purely local and works for all $p$-adic fields and all residual Galois representations.