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On torsion in the cohomology of locally symmetric varieties. (English) Zbl 1345.14031
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 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.

MSC:
14G20 Local ground fields in algebraic geometry
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11S37 Langlands-Weil conjectures, nonabelian class field theory
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
22E50 Representations of Lie and linear algebraic groups over local fields
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