Non-abelian reciprocity laws on a Riemann surface

Details Non-abelian reciprocity laws on a Riemann surface Non-abelian reciprocity laws on a Riemann surface

2010-02-13

arxiv.org/abs/1002.2707v1

Mathematics

Algebraic Geometry

21 pages; Submitted to International Mathematics Research Notices;

Scientific paper

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.

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