Diagrams in the mod \(p\) cohomology of Shimura curves. (English) Zbl 1480.11081
Summary: We prove a local-global compatibility result in the mod \(p\) Langlands program for \(\operatorname{GL}_2(\mathbb{Q}_{p^f})\). 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 \(p\).
MSC:
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 14G35 | Modular and Shimura varieties |
| 11F80 | Galois representations |
| 11S37 | Langlands-Weil conjectures, nonabelian class field theory |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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