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

TL;DR

This paper investigates the detailed structure of mod p cohomology representations of GL_2 over unramified p-adic fields, revealing their subquotient composition, filtrations, invariants, and Hilbert series under certain conditions.

Contribution

It provides refined structural results on subquotients, filtrations, and invariants of mod p cohomology representations, extending previous finiteness results.

Findings

Finite length of representations in certain conditions

Explicit descriptions of Iwahori-socle filtrations

Calculation of Hilbert series for these representations

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.