$3$-rank of ambiguous class groups of cubic Kummer extensions

TL;DR

This paper characterizes cubic Kummer extensions of $Q(zeta_3)$ with specific ambiguous class group structures, investigates their conductors, and classifies fields based on unit group cohomology, combining theoretical and computational methods.

Contribution

It provides a complete characterization of extensions with cyclic 3-group of ambiguous classes of order 3 and analyzes their conductors and unit group cohomology.

Findings

Identified conditions for cyclic 3-groups of ambiguous classes of order 3.

Determined the multiplicity of conductors for these abelian extensions.

Classified fields according to the cohomology of their unit groups.

Abstract

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langle\sigma\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(\sigma)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_{k}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_{k}$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.