Ramification theory of reciprocity sheaves, II, Higher local symbols

TL;DR

This paper develops a formalism for higher local symbols in reciprocity sheaves, connecting residue maps, class field theory, and reciprocity laws, and characterizing moduli via these symbols.

Contribution

It introduces a new construction of higher local symbols along Parsin chains and relates them to existing class field theory and reciprocity laws.

Findings

Constructed higher local symbols along Parsin chains.

Connected higher local symbols to Kato's higher local class field theory.

Characterized the modulus of reciprocity sheaf sections using these symbols.

Abstract

We construct a theory of higher local symbols along Parsin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Parsin-Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato's higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.