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Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

Published online by Cambridge University Press:  30 September 2014

Ivan Horozov  and
Matt Kerr
Ivan Horozov
Affiliation:
Washington University in St Louis, Department of Mathematics, Campus Box 1146, One Brookings Drive, St Louis, MO 63130, USA, [email protected]
Matt Kerr
Affiliation:
Washington University in St Louis, Department of Mathematics, Campus Box 1146, One Brookings Drive, St Louis, MO 63130, USA
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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.

Keywords

reciprocity lawscomplex algebraic surfacesiterated integrals14C3032J2555P35
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 2 , October 2014 , pp. 273 - 312
Copyright
Copyright © ISOPP 2014 

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