Locally analytic vectors in the completed cohomology of unitary Shimura curves
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911174965067776 |
|---|---|
| author | Qiu, Tian Su, Benchao |
| author_facet | Qiu, Tian Su, Benchao |
| contents | 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 $σ$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $σ$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $σ:L\hookrightarrow E$ 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 $σ$-algebraic vectors of the completed cohomology then it is $σ$-de Rham. Finally, we give a geometric realization of some locally $σ$-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 $\mathrm{GL}_2(L)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally analytic vectors in the completed cohomology of unitary Shimura curves Qiu, Tian Su, Benchao Number Theory 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 $σ$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $σ$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $σ:L\hookrightarrow E$ 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 $σ$-algebraic vectors of the completed cohomology then it is $σ$-de Rham. Finally, we give a geometric realization of some locally $σ$-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 $\mathrm{GL}_2(L)$. |
| title | Locally analytic vectors in the completed cohomology of unitary Shimura curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.10290 |