Higher order group cohomology and the Eichler-Shimura map

TL;DR

This paper introduces higher order group cohomology, explores its properties, and constructs an Eichler-Shimura map connecting higher order cusp forms to cohomology groups using modular symbols.

Contribution

It defines higher order group cohomology and constructs a novel Eichler-Shimura homomorphism linking cusp forms to cohomology.

Findings

Higher order group cohomology is well-defined and exhibits key properties.

An Eichler-Shimura map for higher order forms is explicitly constructed.

The work bridges cusp forms and cohomology via modular symbols.

Abstract

Higher order group cohomology is defined and first properties are given. Using modular symbols, an Eichler-Shimura homomorphism is constructed mapping spaces of higher order cusp forms to higher order cohomology groups.