Eichler-Shimura isomorphism and group cohomology on arithmetic groups

TL;DR

This paper provides a cohomological interpretation of the Eichler-Shimura isomorphism for automorphic forms on quaternion algebra groups over totally real fields, linking automorphic representations to group cohomology.

Contribution

It offers a new perspective by interpreting the Eichler-Shimura isomorphism as a connection morphism in a cohomological exact sequence for quaternion algebra groups.

Findings

Cohomological interpretation of Eichler-Shimura isomorphism

Connection morphism in exact sequences of G-modules

Applicable to quaternion algebra groups over totally real fields

Abstract

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