Governing fields for hyperelliptic function fields

TL;DR

This paper investigates the structure of class groups in hyperelliptic function fields, demonstrating that their 8-rank is controlled by splitting conditions in specific governing fields, extending number field results to function fields.

Contribution

It introduces a Rédéi reciprocity law for function fields and establishes the existence of governing fields for hyperelliptic function fields' 8-rank.

Findings

8-rank of class groups is governed by splitting conditions

Existence of governing fields for hyperelliptic function fields

Extension of Rédéi reciprocity law to function fields

Abstract

We study the 8-rank of class groups of hyperelliptic function fields and show that such 8-ranks are governed by splitting conditions in so-called governing fields. A similar result was proven for quadratic number fields by Stevenhagen, who used a theory of R\'edei symbols and R\'edei reciprocity to do so. We introduce a version of the R\'edei reciprocity law for function fields and use this to show existence of governing fields.