Multiple period integrals and cohomology

Bruggeman, Roelof; Choie, Youngju

doi:10.2140/ant.2016.10.645

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Multiple period integrals and cohomology

Roelof W. Bruggeman and Youngju Choie

Vol. 10 (2016), No. 3, 645–664

DOI: 10.2140/ant.2016.10.645

Abstract

We give a version of the Eichler–Shimura isomorphism with a nonabelian $H^{1}$ 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 that 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 nonabelian version of the Eichler–Shimura map.

Keywords

cusp form, iterated integral, noncommutative cohomology

Mathematical Subject Classification 2010

Primary: 11F67

Secondary: 11F75

Milestones

Received: 29 June 2015

Revised: 26 February 2016

Accepted: 28 February 2016

Published: 12 June 2016

Authors

Roelof W. Bruggeman
Mathematisch Instituut Universiteit te Utrecht Postbus 80010 3508 TA Utrecht Netherlands

Youngju Choie
Department of Mathematics and PMI Postech Pohang 790-784 South Korea

Vol. 10, No. 3, 2016