On some $p$-adic and mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$

TL;DR

This paper investigates certain $p$-adic and mod $p$ representations of the quaternion algebra over $Q_p$, establishing finite length properties for a class of admissible Banach space representations of $D^{ imes}$.

Contribution

It proves that specific admissible unitary Banach space representations of $D^{ imes}$ originating from global contexts are topologically of finite length.

Findings

Representations of $D^{ imes}$ have finite length under certain conditions.

The results connect local $p$-adic representations with global origin.

Finite length property aids in understanding the structure of these representations.

Abstract

Let $D$ be the non-split quaternion algebra over $\mathbb{Q}_p$. We prove that a class of admissible unitary Banach space representations of $D^{\times}$ of global origin are topologically of finite length.