Shimura varieties at level and Galois representations

• Published in: Compositio mathematica Vol. 156; no. 6; pp. 1152 - 1230
• Main Authors: Caraiani, Ana, Gulotta, Daniel R., Hsu, Chi-Yun, Johansson, Christian, Mocz, Lucia, Reinecke, Emanuel, Shih, Sheng-Chi
• Format: Journal Article
• Language: English; French
• Published: London Cambridge University Press 01.06.2020
• Subjects: 11F75; 11F80; 11G18; 14G22; 2010 Mathematics Subject Classification; Algebra; Eigenvalues; Homology; Representations
• ISSN: 0010-437X; 1570-5846; 1570-5846
• DOI: 10.1112/S0010437X20007149

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].