Duke Mathematical Journal
- Duke Math. J.
- Volume 161, Number 12 (2012), 2311-2413.
Local-global compatibility and the action of monodromy on nearby cycles
We strengthen the local-global compatibility of Langlands correspondences for in the case when is even and . Let be a CM field, and let be a cuspidal automorphic representation of which is conjugate self-dual. Assume that is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an -adic Galois representation and proved the local-global compatibility up to semisimplification at primes not dividing . We extend this compatibility by showing that the Frobenius semisimplification of the restriction of to the decomposition group at corresponds to the image of via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator on the complex of nearby cycles on a scheme which is locally étale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for as above.
Duke Math. J. Volume 161, Number 12 (2012), 2311-2413.
First available in Project Euclid: 6 September 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F80: Galois representations 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Caraiani, Ana. Local-global compatibility and the action of monodromy on nearby cycles. Duke Math. J. 161 (2012), no. 12, 2311--2413. doi:10.1215/00127094-1723706. https://projecteuclid.org/euclid.dmj/1346936109