# Abelian arithmetic Chern-Simons theory and arithmetic linking numbers

## TL;DR

This paper develops an arithmetic analogue of linking numbers using duality theorems and residue symbols, providing a new perspective on number theory inspired by knot theory concepts.

## Contribution

It introduces a novel formalism connecting arithmetic duality, residue symbols, and linking numbers, bridging knot theory and number theory.

## Key findings

- Defined arithmetic linking numbers and height pairings using duality theorems
- Computed arithmetic linking numbers in terms of n-th power residue symbols
- Established an arithmetic analogue of the path-integral formula for linking numbers

## Abstract

Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power residue symbols. This formalism leads to a precise arithmetic analogue of a 'path-integral formula' for linking numbers.