Shimura varieties at level $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ and Galois representations

We show that the compactly supported cohomology of certain U(n, n)- or Sp(2n)-Shimura varieties with Gamma(1)(p(infinity))-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 [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].