On the Breuil-Schneider conjecture II: Potentially crystalline non-generic case

TL;DR

This paper advances the understanding of the Breuil-Schneider conjecture by proving it in the potentially crystalline non-generic case, especially when all components are automorphic, thus confirming the conjecture unconditionally in these scenarios.

Contribution

It extends previous results to non-smooth points on automorphic components, establishing the conjecture's validity in the potentially crystalline non-generic case.

Findings

Proves the Breuil-Schneider conjecture for non-smooth points on automorphic components.

Shows the conjecture holds unconditionally when all components are automorphic in certain deformation rings.

Improves understanding of the conjecture in the potentially crystalline non-generic setting.

Abstract

This paper improves some results of the author's previous work. We will investigate the case of non-smooth points on an automorphic components and prove Breuil-Schenider conjecture. As a consequence we will see that in case when all the components are automorphic in the potentially crystalline deformation rings, the Breuil-Schenider conjecture holds unconditionally.