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

2012/03/31 by Matthew Emerton, Toby Gee
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Cohomology #Conjecture #Galois module #Geometry and complex manifolds #Mathematics #Modular form #Pure mathematics #Shimura variety #Unitary state #math.NT

paper · pdf · doi:10.2140/ant.2015.9.1035

published as Algebra Number Theory 9 (2015) 1035-1088 · Essentially final version; to appear in Algebra and Number Theory

arxiv created 2015/04/14 · openalex publication_date 2015/06/21 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

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.

Citations

• On the derived category of perverse sheaves
• Introduction to Shimura varieties with bad reduction of parahoric type
• Cohomology of Siegel Varieties with p-adic integral coefficients and Applications
• Modularity of certain potentially Barsotti-Tate Galois representations