Shimura varieties at level and Galois representations

Journal article, 2020

We show that the compactly supported cohomology of certain - or -Shimura varieties with -level vanishes above the middle degree. The only assumption is that we work over a CM field in which the prime splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945-1066; MR 3418533] and Newton-Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].