Triple symbols in arithmetic

TL;DR

This paper introduces a cohomological formulation of a mod n triple symbol in arithmetic, inspired by Chern–Simons theory, and demonstrates its consistency with existing definitions for specific cases.

Contribution

It presents a novel cohomological approach to defining mod n triple symbols in arithmetic, extending previous work and connecting to topological concepts.

Findings

The new symbol aligns with Rédéi's for n=2.

It agrees with Amano–Mizusawa–Morishita's for n=3.

Provides a unified cohomological framework for triple symbols.

Abstract

Triple symbols are arithmetic analogues of the mod $n$ triple linking number in topology, where $n > 1$ is an integer. In this paper, we introduce a cohomological formulation of a mod $n$ triple symbol for characters over a number field containing a primitive $n$-th root of unity. Our definition is motivated by the arithmetic Chern--Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of R\'edei when $n=2$ and of Amano--Mizusawa--Morishita when $n=3$.