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The Eichler-Shimura isomorphism realizes the automorphic representationgenerated by an automorphic newform in certain cohomology of an arithmeticgroup. In this short note, we give a cohomological interpretation of theEichler-Shimura isomorphism as a connection morphism of certain exact sequenceof G-modules, for the multiplicative group G of any quaternion algebra over atotally real field.

Eichler-Shimura isomorphism and group cohomology on arithmetic groups