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 $\Pi$ 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.