Type: Article
Publication Date: 2012-09-06
Citations: 119
DOI: https://doi.org/10.1215/00127094-1723706
We strengthen the local-global compatibility of Langlands correspondences for GLn in the case when n is even and l≠p. 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.