#### Vol. 8, No. 7, 2014

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Monodromy and local-global compatibility for $l=p$

### Ana Caraiani

Vol. 8 (2014), No. 7, 1597–1646
##### 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 $\Pi$ a cuspidal automorphic representation of ${GL}_{n}\left({\mathbb{A}}_{L}\right)$ which is conjugate self-dual and regular algebraic. In this case, there is an $l$-adic Galois representation associated to $\Pi$, 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 $\Pi$ 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.

##### Keywords
Galois representations, automorphic forms, local-global compatibility, monodromy operator, crystalline cohomology
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11G18, 11R39