A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

TL;DR

This paper computes and classifies extension groups in the category of dual $p$-adic Banach space representations of $ ext{GL}_2(Q_p)$, focusing on those from the $p$-adic local Langlands correspondence, and applies these results to cohomological contexts.

Contribution

It provides a complete classification of extensions for certain $p$-adic representations related to the local Langlands correspondence, with applications to étale cohomology.

Findings

Classified extensions for representations from the $p$-adic local Langlands correspondence.

Proved vanishing of extensions between duals of reducible and supercuspidal representations.

Applied extension computations to étale cohomology of Drinfeld spaces.

Abstract

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the \`etale cohomology of the finite level Drinfeld spaces.