Shimura varieties at level Γ_1(p^∞) and Galois representations

We show that the compactly supported cohomology of certain U(n,n) or Sp(2n)-Shimura varieties with Γ_1(p^∞)-level vanishes above the middle degree. The only assumption is that we work over a CM field F in which the prime p splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for GL_n/F. 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 and Newton-Thorne.