See Hannu thesis.
Tame symbols and reciprocity laws on arithmetic surfaces. http://front.math.ucdavis.edu/1203.6712. Author: Dongwen Liu
Artin reciprocity: See Frei in Abel volume for historical background
http://mathoverflow.net/questions/96290/weil-reciprocity-vs-artin-reciprocity
arXiv:1002.4848 A categorical proof of the Parshin reciprocity laws on algebraic surfaces from arXiv Front: math.CT by Denis Osipov, Xinwen Zhu We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.
arXiv:1002.2707 Non-abelian reciprocity laws on a Riemann surface from arXiv Front: math.AG by Ivan Horozov 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.
arXiv:1209.1217 K-groups of reciprocity functors from arXiv Front: math.AG by Florian Ivorra, Kay Rülling In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get closed to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or Kähler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.
arXiv:1206.5817 Reciprocity Laws on Algebraic Surfaces via Iterated Integrals from arXiv Front: math.AG by Ivan Horozov, Matt Kerr In this paper we define a new symbol, called the 4-function symbol, on a complex algebraic surface, which satisfies two types of reciprocity laws. In comparison the Parshin symbol on a surface is defined for 3 non-zero rational functions. Both the 4-function symbol and the Parshin symbol are expressed as a product of more primitive symbols, which we call bi-local symbols. They also satisfy reciprocity laws and occur naturally, when iterated integrals are used. The key technical ingredient is the notion of iterated integrals on membranes. In terms of such integrals, we not only prove reciprocity laws but we give an interpretation of the symbols as parallel transports on the loop space of a variety. Moreover, such integrals give a relation between the 4-function symbol and the Riemann curvature tensor
The appendix contains a K-theoretic variant of the 4-function symbol, which differs by a sign. This difference causes one of the reciprocity laws to fail, suggesting that iterated integrals play an essential role in the definition of the (correct) 4-function symbol.
nLab page on Reciprocity laws