Non-abelian reciprocity laws on a Riemann surface

TL;DR

This paper generalizes classical relations among periods on Riemann surfaces to infinite families of relations involving iterated integrals, leading to new reciprocity laws and applications in noncommutative modular symbols.

Contribution

It introduces a framework for reciprocity laws involving generating series of iterated integrals on Riemann surfaces, extending classical results to higher degrees.

Findings

Derived infinitely many relations from generating series of iterated integrals.

Refined Manin's noncommutative modular symbol to include Eisenstein series.

Provided constructions relevant for multidimensional reciprocity laws.

Abstract

On a Riemann surface there are relations among the periods of holomorphic differential forms, called Riemann's relations. If one looks carefully in Riemann's proof, one notices that he uses iterated integrals. What I have done in this paper is to generalize these relations to relations among generating series of iterated integrals. Since the main result is formulated in terms of generating series, it gives infinitely many relations - one for each coefficient of the generating series. The lower order terms give the well known classical relations. The new result is reciprocity for the higher degree terms, which give non-trivial relations among iterated integrals on a Riemann surface. As an application we refine the definition of Manin's noncommutative modular symbol in order to include Eisenstein series. Finally, we have to point out that this paper contains some constructions needed for multidimensional reciprocity laws like a refinement of one of the Kato-Parshin reciprocity laws.