Daniel Gulotta (Columbia University)
Vanishing of cohomology for some Shimura varieties at level \Gamma_1(p^{\infty})

Abstract: Let F be a totally real or CM field. Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n F. The construction makes use of symplectic or unitary Shimura varieties. I will show that when F is totally split at p, the compactly supported cohomology of these Shimura varieties at level \Gamma_1(p^{\infty}) vanishes above the middle degree. The proof involves the study of the Hodge-Tate period map and the Bruhat decomposition on the flag variety. I will also mention an application of this result to Galois representations. This is joint work in progress with A. Caraiani, C.Y. Hsu, C. Johansson, L. Mocz, E. Reinecke, and S.C. Shih.