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.
"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