Monodromy and local-global compatibility for l=p

Abstract

We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane's weight spectral sequence for log crystalline cohomology.

34 pages

References in corpus (1)

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

Cited by in corpus (8)

• Une interprétation modulaire de la variété trianguline
• Deformations of polarized automorphic Galois representations and adjoint Selmer groups
• A local model for the trianguline variety and applications
• On the Breuil-Schneider conjecture: Generic case
• Automorphic L-invariants for reductive groups
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• Smoothness of definite unitary eigenvarieties at critical points
• Globally realizable components of local deformation rings