On the distribution of shapes of pure quartic number fields

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

, the shape is a point in the space of shapes

script upper S n minus 1 $\mathcal{S}_{n-1}$ 𝒮 n − 1

, which is the double quotient

GLn minus 1 left parenthesis double struck upper Z right parenthesis backslash GLn minus 1 left parenthesis double struck upper R right parenthesis slash GOn minus 1 left parenthesis double struck upper R right parenthesis $\operatorname{GL}_{n-1}(\mathbb{Z}) \backslash \operatorname{GL}_{n-1}(\mathbb{R}) / \operatorname{GO}_{n-1}(\mathbb{R})$ GL n − 1 ( ℤ ) \ GL n − 1 ( ℝ ) / GO n − 1 ( ℝ )

. In this paper, we investigate the distribution of shapes in the family of pure quartic fields

Km equals double struck upper Q left parenthesis m 4 right parenthesis $K_m = \mathbb{Q}(\sqrt[4]{m})$ K m = ℚ ( m 4 )

. We prove that the shape of

Km $K_m$ K m

lies on one of

5 $5$ 5

explicitly described torus orbits in

script upper S 3 $\mathcal{S}_3$ 𝒮 3

, determined by the sign and residue class of

mmod 32 $m \bmod 32$ m mod 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

script upper S 3 $\mathcal{S}_3$ 𝒮 3

induced by the Haar measure on

GL 3 left parenthesis double struck upper R right parenthesis $\operatorname{GL}_3(\mathbb{R})$ GL 3 ( ℝ )

.