A General Reciprocity Law for Symbols on Arbitrary Vector Spaces

TL;DR

This paper develops a comprehensive reciprocity law framework for symbols on any vector space, unifying classical and recent reciprocity laws under a single general theory.

Contribution

It introduces a universal reciprocity law for symbols on arbitrary vector spaces, encompassing classical and recent specific reciprocity laws as special cases.

Findings

Classical reciprocity laws are special cases of the new theory

Several recent reciprocity laws can be derived from the general framework

Provides a unified approach to reciprocity laws in algebraic geometry

Abstract

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces, and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov - M. Kerr and the author -together with D. Hern\'andez Serrano-, can also be deduced from this general expression.