Duke Mathematical Journal

Local-global compatibility and the action of monodromy on nearby cycles

Ana Caraiani

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Abstract

We strengthen the local-global compatibility of Langlands correspondences for $\operatorname{GL}_{n}$ in the case when $n$ is even and $l\neq p$. Let $L$ be a CM field, and let $\Pi$ be a cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $\Pi_{\infty}$ is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(\Pi)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of $R_{l}(\Pi)$ to the decomposition group at $v$ corresponds to the image of $\Pi_{v}$ via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that $\Pi$ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator $N$ 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 $\Pi$ as above.

Article information

Source
Duke Math. J. Volume 161, Number 12 (2012), 2311-2413.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1346936109

Digital Object Identifier
doi:10.1215/00127094-1723706

Mathematical Reviews number (MathSciNet)
MR2972460

Zentralblatt MATH identifier
06095601

Subjects
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]

Citation

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. http://projecteuclid.org/euclid.dmj/1346936109.


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