A Hilbert reciprocity law on 3-manifolds

TL;DR

This paper develops an analogue of the Hilbert reciprocity law within the context of 3-manifolds, using homological idelic class field theory to relate knot theory and number theory concepts.

Contribution

It introduces a novel formulation of the Hilbert reciprocity law for rational homology 3-spheres with infinite links, bridging topology and arithmetic through intersection forms.

Findings

Formulation of a Hilbert reciprocity law on 3-manifolds.

Analogies between intersection forms and Hilbert symbols.

Cyclic covers of links as analogues of Kummer extensions.

Abstract

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.