[OS Arithm. Geom. and Reps Th.] Galois representations occurring in the cohomology of locally symmetric spaces

By a theorem of Scholze there exist Galois representations valued in Hecke algebras associated with the cohomology of locally symmetric spaces. The characterizing property of these representations is a compatibility between the action of Frobenius and the action of certain Hecke operators. In joint work with Hevesi, Thorne and Whitmore we prove that the Galois representations in question are also compatible with the local Langlands correspondence at p-adic places, up to semisimplification. The novelty of our work is that we impose no
assumptions on the residual Galois representations. As an application we can prove new cases of vanishing of adjoint Bloch-Kato Selmer groups associated with cuspidal, cohomological automorphic representations.