The study reveals how class groups' 8-ranks in hyperelliptic function fields relate to governing fields, suggesting connections to quadratic number fields.
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.
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Joppe Stokvis (2025) studied this question.
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