Diagrams in the mod $p$ cohomology of Shimura curves

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 $(\phi ,\Gamma )$-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 $p$.

60 pages, minor changes and fixes

References in corpus (3)

• Multiplicity one for wildly ramified representations
• Multiplicity one for the mod $p$ cohomology of Shimura curves: the tame case
• On mod $p$ local-global compatibility for ${\mathrm{GL}}_n({\mathbf{Q}}_p)$ in the ordinary case

Cited by in corpus (5)

• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• On irreducible supersingular representations of ${\mathrm{GL}}_2(F)$
• Serre weights for three-dimensional wildly ramified Galois representations
• On the constituents of the mod $p$ cohomology of Shimura curves
• On the mod $p$ cohomology for ${\mathrm{GL}}_2$: the non-semisimple case