Monodromy and local-global compatibility for l = p

Caraiani, Ana

doi:10.2140/ant.2014.8.1597

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

Ana Caraiani

Vol. 8 (2014), No. 7, 1597–1646

DOI: 10.2140/ant.2014.8.1597

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 ${GL}_{n} (A_{L})$ 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.

Keywords

Galois representations, automorphic forms, local-global compatibility, monodromy operator, crystalline cohomology

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11G18, 11R39

Milestones

Received: 27 April 2013

Revised: 31 March 2014

Accepted: 18 May 2014

Published: 21 October 2014

Authors

Ana Caraiani
Department of Mathematics Princeton University Fine Hall Washington Road Princeton, NJ 08544 United States

Vol. 8, No. 7, 2014