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

By Ana Caraiani. 2010

143 citations recorded by the source corpus.

We strengthen the local-global compatibility of Langlands correspondences for GLn in the case when n is even and l 6D p. Let L be a CM field, and let ... be a cuspidal automorphic representation of GLn.AL/ which is conjugate self-dual. Assume that...1 is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representationRl..../ 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..../ to the decomposition group at v corresponds to the image of ...v 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 N on the complex of nearby cycles on a scheme which is locally etale 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a33dc1bf06bc13b729b748282c14b52121e7a9ce

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Citations in the corpus

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Weight cycling and Serre-type conjectures for unitary groups (2011, result): “3 of [Lab09] and the main results of [Car10] and [BLGGT11].”
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Perfectoid Spaces (2011, background): “For this reason, we recall some necessary background in the next section.”
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Potential automorphy and change of weight (2010, methodology): “any people (we mention in particular Bellaiche, Caraiani, Chenevier, Clozel, Harris, Kottwitz, Labesse, Shin and R.T., and the references [Kot92], [Clo90], [HT01], [BC11], [Shi11], [CHLN11], [Clo10], [Car12a] and [Car12b]) we can associate l-adic representations to regular algebraic cuspidal polariz”