April 16, 2020
Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J.