On the finite length of some $p$-adic representations of the quaternion algebra over $\Q_p$

TL;DR

This paper proves that for $p\ge5$, a class of admissible unitary Banach space representations of $D^{\times}$ arising from the $p$-adic local Langlands program are topologically of finite length, even when global origin assumptions are removed and some non-generic residual cases are allowed. The authors combine Scholze's functor for $\mathrm{GL}_2(\mathbb{Q}_p)$ with Colmez's Montréal functor, and apply Taylor–Wiles–Kisin patching to relate local Galois representations to global automorphic data via completed cohomology and deformation rings, establishing an $\rho$-typic decomposition $\check{\mathcal S}^1(\Pi(\rho))\cong \rho\boxtimes {\rm JL}(\rho)$. A refined finiteness criterion controls multiplicities in the quaternionic setting, and the locally algebraic vector spaces are shown to be finite dimensional in the key cases. The results broaden the scope of the $p$-adic Langlands program for quaternion algebras by providing concrete finite-length structure results and clarifying the interaction between patched global data and local representations.

Abstract

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

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• Definition 2.2
• Theorem 3.1
• proof