Monodromy and local-global compatibility for l=p

TL;DR

This paper enhances the understanding of the local-global compatibility in the Langlands program for GL_{n} when n is even and l=p, by extending compatibility to Frobenius semisimplification through monodromy operator analysis.

Contribution

It generalizes the compatibility between local and global Langlands correspondences for GL_{n} with l=p to all cases by identifying the monodromy operator, beyond previous semisimplification results.

Findings

Extended compatibility to Frobenius semisimplification for all cases.

Derived a generalization of Mokrane's weight spectral sequence.

Identified the monodromy operator on the global side.

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}(\mathbb{A}_{L}) 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.