NDLI: Diagrams in the mod p cohomology of Shimura curves

Content Provider	Scilit
Author	Dotto, Andrea Le, Daniel
Copyright Year	2021
Description	Journal: Compositio Mathematica 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 $p$ .
Ending Page	1723
Starting Page	1653
ISSN	00221295
e-ISSN	15705846
DOI	10.1112/s0010437x21007375
Journal	Compositio Mathematica
Issue Number	8
Volume Number	157
Language	English
Publisher	Wiley-Blackwell
Publisher Date	2021-08-01
Access Restriction	Open
Subject Keyword	Journal: Compositio Mathematica Shimura Curves
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory

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Diagrams in the mod p cohomology of Shimura curves