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Diagrams in the mod p cohomology of Shimura curves

Part of: Lie groups Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 July 2021

Andrea Dotto  and
Daniel Le
Andrea Dotto
Affiliation:
University of Chicago, 5734 South University Avenue, Chicago, IL60637, [email protected]
Daniel Le
Affiliation:
Purdue University, 150 North University Street, West Lafayette, IN47907, [email protected]
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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 $p$.

Keywords

Galois representationsmod p Langlands program

MSC classification

Primary: 11F80: Galois representations
Secondary: 11S37: Langlands-Weil conjectures, nonabelian class field theory 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 8 , August 2021 , pp. 1653 - 1723
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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