I will prove that the compactly supported cohomology of certain unitary Shimura varieties at level Gamma_1(p^\infty)l vanishes above the middle degree. I will describe the steps in the proof using modular curves as a toy model. I will also discuss an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in progress with A. Dotto, D. Gulotta, M. Gerbelli-Gauthier, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, M. Santos, and S.C. Shih.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/246