On the distribution of shapes of totally real multiquadratic number fields

TL;DR

This paper investigates how the geometric shapes of totally real multiquadratic number fields distribute within a specific mathematical space, confirming a conjecture about their distribution pattern.

Contribution

It proves that the shape distribution of these number fields is determined by a particular torus orbit, resolving a conjecture of Haidar.

Findings

Distribution governed by measure restriction to a torus orbit

Confirmed conjecture of Haidar on shape distribution

Provides new insights into the geometry of number fields

Abstract

The shape of a number field $K$ of degree $m$ 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 $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. The double quotient space is equipped with a natural measure $\mu$ which is induced from the Haar measure on $\mathrm{GL}_{m-1}(\mathbb R)$. We study the distribution of shapes of totally real multiquadratic number fields of degree $m:=2^n$ in which $2$ is unramified. We show that the distribution is governed by the restriction of $\mu$ to a certain torus orbit in $\mathscr{S}_{m-1}$. Our result resolves a conjecture of Haidar.