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arXiv · 2506.16293

On the constituents of the mod $p$ cohomology of Shimura curves

Christophe Breuil·Florian Herzig·Yongquan Hu·Stefano Morra·Benjamin Schraen
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Abstract

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 $\pi$ 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 $\pi$, 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 $\pi$ as Iwahori-representation under these conditions.

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Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen. 2025-06-19. On the constituents of the mod $p$ cohomology of Shimura curves. https://doi.org/10.5802/jep.346

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