Eichler–Shimura isomorphism and group cohomology on arithmetic groups

Abstract

In this article, we give a group cohomological interpretation to the Eichler–Shimura isomorphism. For any quaternion algebra A over a totally real field with multiplicative group G, we interpret a weight (k1,k2,⋯,kd)-automorphic form of G as a G(F)-invariant homomorphism of (G∞,K∞)-modules. Then the Eichler–Shimura isomorphism is given by the connection morphism provided by the natural exact sequences defining the (G∞,K∞)-module of discrete series of weight (k1,k2,⋯,kd).