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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 vanishesabove 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 toGalois representations for torsion in the cohomology of the locally symmetricspaces for $\mathrm{GL}_n/F$. More precisely, we use the vanishing result forShimura varieties to eliminate the nilpotent ideal in the construction of theseGalois representations. This strengthens recent results of Scholze andNewton-Thorne.

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