Vol. 9, No. 5, 2015

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$p$-adic Hodge-theoretic properties of étale cohomology with mod $p$ coefficients, and the cohomology of Shimura varieties

Matthew Emerton and Toby Gee

Vol. 9 (2015), No. 5, 1035–1088
Abstract

We prove vanishing results for the cohomology of unitary Shimura varieties with integral coefficients at arbitrary level, and deduce applications to the weight part of Serre’s conjecture. In order to do this, we show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of p, embeds into the reduction modulo p of a semistable Galois representation with Hodge–Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree greater than 1).

Keywords
p-adic Hodge theory, Shimura varieties
Mathematical Subject Classification 2010
Primary: 11F33
Milestones
Received: 24 October 2013
Revised: 13 March 2015
Accepted: 10 April 2015
Published: 21 June 2015
Authors
Matthew Emerton
Mathematics Department
University of Chicago
Chicago, IL 60637
United States
Toby Gee
Imperial College London
London SW7 2AZ
United Kingdom