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 $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. 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.