Reciprocity Laws for the Higher Tame Symbol and the Witt Symbol on an Algebraic Surface

TL;DR

This paper establishes reciprocity laws for a combined symbol derived from Parshin's higher Witt pairing and the higher tame pairing on an arithmetic surface, extending classical reciprocity concepts to higher-dimensional algebraic geometry.

Contribution

It introduces and proves reciprocity laws for a new symbol combining the higher Witt and tame pairings on algebraic surfaces, using advanced techniques from Morrow and Romo.

Findings

Proves reciprocity laws for the combined symbol on algebraic surfaces.

Extends classical reciprocity to higher-dimensional algebraic geometry.

Utilizes techniques of Morrow and Romo for the proofs.

Abstract

Parshin's higher Witt pairing on an arithmetic surface can be combined with the higher tame pairing to form a symbol taking values in the absolute abelian Galois group of the function field. We prove reciprocity laws for this symbol using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol.