A refinement of the Parshin symbol for surfaces

TL;DR

This paper introduces a refined Parshin symbol for algebraic surfaces using Chen's iterated integrals, along with a logarithmic version, and establishes reciprocity laws for both.

Contribution

It provides a novel refinement of the Parshin symbol for surfaces and constructs a logarithmic version, extending the algebraic reciprocity laws.

Findings

Refined Parshin symbol recovers the original when cyclically permuted.

Constructed a logarithmic version of the Parshin symbol.

Proved reciprocity laws for the new symbols.

Abstract

On an algebraic curve there are Tate symbols, which satisfy Weil reciprocity law. The analogues in higher dimensions are the Parshin symbols, which satisfy Kato-Parshin reciprocity laws. We give a refinement of the Parshin symbol for surfaces, using iterated integrals in the sense of Chen. The product of the refined symbol over the cyclic permutations of the functions recovers the Parshin symbol. Also, we construct a logarithmic version of the Parshin symbol. We prove reciprocity laws for both the refined symbol and a logarithm of the Parshin symbol.