15 September 2012 Local-global compatibility and the action of monodromy on nearby cycles
Ana Caraiani
Duke Math. J. 161(12): 2311-2413 (15 September 2012). DOI: 10.1215/00127094-1723706


We strengthen the local-global compatibility of Langlands correspondences for GLn in the case when n is even and lp. Let L be a CM field, and let Π be a cuspidal automorphic representation of GLn(AL) which is conjugate self-dual. Assume that Π is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representation Rl(Π) 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 Rl(Π) to the decomposition group at v corresponds to the image of Πv via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that Π 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 Π as above.


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Ana Caraiani. "Local-global compatibility and the action of monodromy on nearby cycles." Duke Math. J. 161 (12) 2311 - 2413, 15 September 2012. https://doi.org/10.1215/00127094-1723706


Published: 15 September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 06095601
MathSciNet: MR2972460
Digital Object Identifier: 10.1215/00127094-1723706

Primary: 11R39
Secondary: 11F70 , 11F80 , 14G35

Rights: Copyright © 2012 Duke University Press


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Vol.161 • No. 12 • 15 September 2012
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