Shimura varieties at level \(\Gamma_1(p^\infty)\) and Galois representations

We show that the compactly supported cohomology of certain \(\mathrm{U}(n,n)\) or \(\mathrm{Sp}(2n)\)-Shimura varieties with \(\Gamma_1(p^\infty)\)-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 \(\mathrm{GL}_n/F\). 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.