On the distribution of shapes of pure quartic number fields

TL;DR

This paper studies the geometric shapes of pure quartic number fields, revealing they lie on specific torus orbits and their distribution combines continuous and discrete measures, addressing a question by Bhargava and H.

Contribution

It explicitly describes the shape distribution of pure quartic fields, showing they lie on ten torus orbits and are governed by mixed measures, providing new insights into number field shape distributions.

Findings

Shapes lie on ten explicit torus orbits in the shape space.

Shape distribution is a product of continuous and discrete measures.

The limiting distribution differs from the natural Haar measure on the shape space.

Abstract

The shape of a number field is a subtle arithmetic invariant arising from the geometry of numbers. It 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. For a number field of degree $n$, the shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $GL_{n-1}(\mathbb{Z}) \backslash GL_{n-1}(\mathbb{R}) / GO_{n-1}(\mathbb{R})$. In this paper, we investigate the distribution of shapes in the family of pure quartic fields $K_m = \mathbb{Q}(\sqrt[4]{m})$. We prove that the shape of $K_m$ lies on one of ten explicitly described torus orbits in $\mathscr{S}_3$, determined by the sign and residue class of $m \bmod 32$. It is shown that the shape on a given torus orbit is completely determined by two parameters, one of which varies continuously, while the other takes values in a discrete set. As a result, the distribution of shapes in this family is governed by a product of a continuous and a discrete measure. Our results shed new light on a question posed by Manjul Bhargava and Piper H concerning the distribution of shapes in families of non-generic number fields of fixed degree. Notably, the limiting distribution in our case does not arise as the restriction of the natural measure on $\mathscr{S}_3$ induced by Haar measure on $GL_3(\mathbb{R})$.