On the distribution of shapes of totally real multiquadratic 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 ${\mathcal{S}}_{m-1}={\mathrm{GL}}_{m-1}(\mathbb{Z})\backslash {\mathrm{GL}}_{m-1}(\mathbb{R})/{\mathrm{GO}}_{m-1}(\mathbb{R})$. 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 ${\mathcal{S}}_{m-1}$. Our result resolves a conjecture of Haidar.

v1: 26 pages