On the distribution of shapes of octic Kummer extensions

TL;DR

This paper investigates the distribution of shapes of octic Kummer extensions, providing an explicit asymptotic formula for their distribution based on shape parameters and discriminant ordering.

Contribution

It introduces a parametrization of shapes via explicit invariants and derives an asymptotic distribution formula for these shapes in octic Kummer extensions.

Findings

Derived an explicit measure for the limiting distribution of shapes.

Established an asymptotic formula ordered by absolute discriminant.

Identified the distribution as a product of 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 under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{n-1} = \mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$. In this paper, we study the distribution of shapes of octic Kummer extensions $L=\mathbb{Q}(i,\sqrt[4]{m})$, where $m\in\mathbb{Z}[i]$ is fourth-power-free. We parametrize these shapes by explicit invariants known as shape parameters and establish an asymptotic formula for their joint distribution ordered by absolute discriminant. The limiting distribution is given by an explicit measure that factors as the product of a continuous measure and a discrete measure arising from local arithmetic conditions.