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

Emerton, Matthew; Gee, Toby

doi:10.2140/ant.2015.9.1035

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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

DOI: 10.2140/ant.2015.9.1035

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

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

Vol. 9, No. 5, 2015