Ordinary local representations and $\Ext$ groups

TL;DR

This paper investigates the properties of certain local representations associated with Galois representations via the $p$-adic local Langlands correspondence, focusing on vanishing results for Ext functors when the representations are ordinary.

Contribution

It establishes new local and global vanishing results for Ext groups involving ordinary representations in the context of the $p$-adic local Langlands correspondence.

Findings

Proves vanishing of Ext groups locally for ordinary representations.

Establishes global vanishing results for Ext functors.

Enhances understanding of the structure of local Galois representations.

Abstract

We can associate an admissible unitary representation $\Pi(\rho_p)$ of $\GL_2(\Q_p)$ with every local Galois representation $\rho_p$ by the $p$-adic local Langlands correspondence. If $\rho_p$ is ordinary, we prove local and global vanishing results for $\Ext$ functors with respect to these representations.