Siegel modular varieties and Borel-Serre compactification of modular curves

To motivate the study of Siegel modular varieties and Borel-Serre compactifications let me recall Scholze’s theorem on torsion classes. We start from an Hecke eigenclass h ∈ H (Xg/Γ,Fp) where Xg is the locally symmetric domain for GLg, and we want to attach to it a continuous semisimple Galois representation ρh : GQ → GLg(Fp) such that the characteristic polynomials of the Frobenius classes at unramified primes are determined by the Hecke eigenvalues of h. Problem: Xg/Γ is not algebraic in general, in fact it could be a real manifold of odd dimension; while the most powerful way we know to construct Galois representations is by considering étale cohomology of algebraic varieties. Idea(Clozel): Find the cohomology of Xg/Γ in the cohomology of the boundary of the Borel-Serre compactification A BS g of Siegel modular varieties.