On the distribution of shapes of sextic pure number fields

TL;DR

This paper studies the distribution of shapes of pure sextic number fields, showing they are equidistributed along certain orbits in shape space, with explicit descriptions and measures for each type.

Contribution

It provides an explicit classification and description of shapes for pure sextic fields, and proves their equidistribution in shape space for each local type.

Findings

Shapes are explicitly described for each type of pure sextic fields.

Shapes are shown to be equidistributed along translated torus orbits.

The limiting distribution combines continuous and discrete measures.

Abstract

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure.