#### 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 ${ℚ}_{p}$, 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
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