Locally analytic vectors in the completed cohomology of unitary Shimura curves

TL;DR

This paper investigates locally analytic vectors in the completed cohomology of unitary Shimura curves, proving classicality and de Rham properties of certain Galois representations, and providing geometric realizations relevant to the $p$-adic local Langlands program.

Contribution

It introduces new methods to analyze locally analytic vectors in completed cohomology, establishing classicality, de Rham conditions, and geometric realizations with applications to the $p$-adic Langlands correspondence.

Findings

Proves classicality for two-dimensional regular $\sigma$-de Rham Galois representations.

Shows that locally $\sigma$-algebraic vectors correspond to $\sigma$-de Rham representations.

Provides geometric realizations of locally $\sigma$-analytic representations of $ ext{GL}_2(L)$.

Abstract

We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular $\sigma$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $\sigma$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $\sigma:L\hookrightarrow E$ is an embedding of $L$ into a sufficiently large finite extension $E$ of $\mathbb{Q}_p$. We also prove that if a two-dimensional representation of $\text{Gal}(\bar L/L)$ appears in the locally $\sigma$-algebraic vectors of the completed cohomology then it is $\sigma$-de Rham. Finally, we give a geometric realization of some locally $\sigma$-analytic representations of $\mathrm{GL}_2(L)$. This realization has some applications to the $p$-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext$^1$-conjecture for $\mathrm{GL}_2(L)$.