Diagrams in the mod $p$ cohomology of Shimura curves

TL;DR

This paper establishes a local-global compatibility in the mod p Langlands program for GL2 over p-adic fields, linking diagrams in cohomology to Galois representations and their restrictions.

Contribution

It proves that certain diagrams in completed cohomology are determined by local Galois restrictions, advancing understanding of the mod p Langlands correspondence for GL2.

Findings

Diagram in cohomology determined by Galois restrictions

Semisimple restrictions lead to tensor inductions of dual Galois representations

Results apply to sufficiently generic residual representations

Abstract

We prove a local-global compatibility result in the mod $p$ Langlands program for $\mathrm{GL}_2(\mathbf{Q}_{p^f})$. Namely, given a global residual representation $\bar{r}$ that is sufficiently generic at $p$, we prove that the diagram occurring in the corresponding Hecke eigenspace of completed cohomology is determined by the restrictions of $\bar{r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi,\Gamma)$-modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar{r}$ to decomposition groups at $p$.