Diagrams in the mod p cohomology of Shimura curves. (7th August 2021)

• Record Type: Journal Article
• Authors: Dotto, Andrea; Le, Daniel
• Is Part Of: Compositio mathematica. Volume 157:Number 8(2021)
• Page Start: 1653
• Page End: 1723
• Publication Date: 2021-08-07
• Subjects: Galois representations -- mod p Langlands program; 11F80 -- 11S37 -- 22E50; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007375
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $p$ .