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

Matthew Emerton; Toby Gee

doi:10.2140/ant.2015.9.1035

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arXiv · 1203.4963

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

Matthew Emerton·Toby Gee

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Abstract

We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, 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 > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serre's conjecture.

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Matthew Emerton, Toby Gee. 2015-04-14. p-adic Hodge-theoretic properties of étale cohomology with mod p coefficients, and the cohomology of Shimura varieties. https://doi.org/10.2140/ant.2015.9.1035

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