Unramified Heisenberg group extensions of number fields

TL;DR

This paper constructs special unramified Heisenberg group extensions over number fields, producing infinite families of quadratic extensions of cyclotomic fields with unramified Galois extensions, advancing understanding of number field extensions.

Contribution

It introduces a method to construct unramified generalized Heisenberg group covers of hyperelliptic curves over number fields, leading to new infinite families of unramified Galois extensions.

Findings

Infinite families of unramified Galois extensions of cyclotomic fields constructed

Generalized Heisenberg group covers used to produce these extensions

Extensions are quadratic and unramified everywhere

Abstract

We construct \'etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized Heisenberg Galois extensions.