Reciprocity Laws on Algebraic Surfaces via Iterated Integrals. Issue 2 (October 2014)

• Record Type: Journal Article
• Authors: Horozov, Ivan; Kerr, Matt
• Is Part Of: Journal of K-Theory. Volume 14:Issue 2:Part 1(2014)
• Page Start: 273
• Page End: 312
• Publication Date: 2014-10
• Subjects: reciprocity laws, -- complex algebraic surfaces, -- iterated integrals; 14C30, -- 32J25, -- 55P35; K-theory -- Periodicals; 512.6605
• DOI: 10.1017/is014006014jkt271
• Languages: English
• ISSNs: 1865-2433

Abstract

Abstract: In this paper we introduce new local symbols, which we call 4-function local symbols. We formulate reciprocity laws for them. These reciprocity laws are proven using a new method - multidimensional iterated integrals. Besides providing reciprocity laws for the new 4-function local symbols, the same method works for proving reciprocity laws for the Parshin symbol. Both the new 4-function local symbols and the Parshin symbol can be expressed as a finite product of newly defined bi-local symbols, each of which satisfies a reciprocity law. The K -theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and the K -theoretic variant of the 4-function local symbol satisfy reciprocity laws, whose proof is based on Milnor K -theory (see the Appendix). The relation of the 4-function local symbols to the double free loop space of the surface is given by iterated integrals over membranes.