Vol. 8, No. 7, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 GLn when n is even and l = p. Let L be a CM field and Π a cuspidal automorphic representation of GLn(AL) 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