Shimura varieties at level and Galois representations - Caraiani, Ana; Gulotta, Daniel R.; Hsu, Chi-Yun; Johansson, Christian; Mocz, Lucia; Reinecke, Emanuel; Shih, Sheng-Chi (Compositio mathematica, 2020)
We show that the compactly supported cohomology of certain $\text{U}(n,n)$ - or $\text{Sp}(2n)$ -Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{∞ })$ -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].