We show that the compactly supported cohomology of certain $\text{U}(n,n)$ - or $\text{Sp}(2n)$ -Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ -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 $\text{GL}_{n}/F$ . 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].

中文翻译：

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Shimura 级变体和 Galois 表示

我们证明了某些 $\text{U}(n,n)$ - 或 $\text{Sp}(2n)$ -Shimura 变体与 $\unicode[STIX]{x1D6E4}_{1} 的紧密支持的上同调(p^{\infty })$ -level 在中度以上消失。唯一的假设是我们在 CM 字段 $F$ 上工作，其中质数 $p$ 完全分裂。我们还在 $\text{GL}_{n}/F$ 的局部对称空间的上同调中给出了扭转的伽罗瓦表示的应用。更准确地说，我们使用 Shimura 变体的消失结果来消除这些伽罗瓦表示构造中的幂零理想。这加强了 Scholze [关于局部对称变体上同调中的扭转，安。的数学。(2) 182 (2015), 945–1066；MR 3418533] 和 Newton-Thorne [CM 场上的 Torsion Galois 表示和派生类别中的 Hecke 代数，论坛数学。西格玛 4 (2016), e21; 先生 3528275]。