Diagrams in the mod p cohomology of Shimura curves. (7th August 2021) - Dotto, Andrea; Le, Daniel
Abstract: We prove a local–global compatibility result in the mod p Langlands program for GL_2(Q_p^f) . Namely, given a global residual representation r appearing in the mod p cohomology of a Shimura curve that is sufficiently generic at p and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod p completed cohomology is determined by the restrictions of r to decomposition groups at p . If these restrictions are moreover semisimple, we show that the (φ, Γ ) -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of r to decomposition groups at p .