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
$\bar {r}$
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
$\bar {r}$
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
$p$
.
Patching and multiplicity 2k for Shimura
curves
