Multiple period integrals and cohomology

TL;DR

This paper extends the Eichler-Shimura isomorphism to a non-abelian setting using iterated integrals, establishing a bijective map that generalizes classical cohomological correspondences.

Contribution

It constructs a non-abelian version of the Eichler-Shimura map by extending Manin's approach, providing a bijective correspondence in non-abelian group cohomology.

Findings

Manin's map is injective but not surjective.

Extension of Manin's map yields a bijective non-abelian Eichler-Shimura map.

Establishes a non-abelian cohomological framework for modular forms.

Abstract

This work gives a version of the Eichler-Shimura isomorphism with a non-abelian $H^1$ in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show Manin's map is injective but far from surjective. By extending Manin's map we are able to construct a bijective map and remarkably this establishes the existence of a non-abelian version of the Eichler-Shimura map.