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

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 ${\mathbb{Q}}_p$, 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 $1$).