Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

TL;DR

This paper introduces a novel method using iterated integrals to prove reciprocity laws on algebraic surfaces, establishing new local symbols and refining existing ones with proven reciprocity properties.

Contribution

It develops a new approach via iterated integrals to prove reciprocity laws, introduces 4-function local symbols, and refines Parshin symbols with bi-local symbols.

Findings

Proved reciprocity laws for Parshin and new 4-function local symbols.

Established bi-local symbols satisfying reciprocity laws.

Defined a K-theoretic variant of the 4-function local symbol.

Abstract

This paper presents a proof of reciprocity laws for the Parshin symbol and for two new local symbols, defined here, which we call 4-function local symbols. The reciprocity laws for the Parshin symbol are proven using a new method - via iterated integrals. The usefulness of this method is shown by two facts - first, by establishing new local symbols - the 4-function local symbols and their reciprocity laws and, second, by providing refinements of the Parshin symbol in terms of 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.