On torsion in the cohomology of locally symmetric varieties
The paper constructs Galois representations associated to Hecke eigenclasses in the torsion cohomology of \(\mathrm{GL}_n\). This holds over a field which is either totally real or CM, assuming conjectures of \textit{J. Arthur} [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)], and without these for CM fields containing an imaginary quadratic field. For the proof the classes are realised in the boundary of Siegel spaces. For these an innovative new technique is used: The infinite cover defined by level-\(p\)-structures of all orders is a perfectoid space which extends to the minimal compactification and admits a Hodge-Tate period map to the dual compact space. Its singular cohomology is described by coherent cohomology, and Hecke eigenclasses can be lifted to classes which are not torsion. For these the machinery of the trace formula allows to construct Galois representations.
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