On the constituents of the mod $p$ cohomology of Shimura curves
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918065509236736 |
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| author | Breuil, Christophe Herzig, Florian Hu, Yongquan Morra, Stefano Schraen, Benjamin |
| author_facet | Breuil, Christophe Herzig, Florian Hu, Yongquan Morra, Stefano Schraen, Benjamin |
| contents | Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$.
When $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we proved in our previous work that the admissible smooth representations $π$ of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length.
In this paper we obtain various refined results about the structure of subquotients of $π$, such as their Iwahori-socle filtrations and $K_1$-invariants, where $K_1$ is the principal congruence subgroup of $\mathrm{GL}_2(\mathcal{O}_K)$.
We also determine the Hilbert series of $π$ as Iwahori-representation under these conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16293 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the constituents of the mod $p$ cohomology of Shimura curves Breuil, Christophe Herzig, Florian Hu, Yongquan Morra, Stefano Schraen, Benjamin Number Theory Representation Theory Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. When $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we proved in our previous work that the admissible smooth representations $π$ of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. In this paper we obtain various refined results about the structure of subquotients of $π$, such as their Iwahori-socle filtrations and $K_1$-invariants, where $K_1$ is the principal congruence subgroup of $\mathrm{GL}_2(\mathcal{O}_K)$. We also determine the Hilbert series of $π$ as Iwahori-representation under these conditions. |
| title | On the constituents of the mod $p$ cohomology of Shimura curves |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2506.16293 |